# New guidelines for the application of Stokes' models to the sinking velocity of marine aggregates

E. C. Laurenceau-Cornec, Frédéric A.C. Le Moigne, Morgane Gallinari, Brivaëla Moriceau, Jordan Toullec, Morten H. Iversen, Anja Engel, Christina L. de La Rocha

► To cite this version:

E. C. Laurenceau-Cornec, Frédéric A.C. Le Moigne, Morgane Gallinari, Brivaëla Moriceau, Jordan Toullec, et al.. New guidelines for the application of Stokes' models to the sinking velocity of marine aggregates. Limnology and Oceanography, 2020, 65 (6), pp.1264-1285. 10.1002/lno.11388. hal-02391799

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# New guidelines for the application of Stokes' models to the sinking velocity of marine aggregates

Emmanuel C. Laurenceau-Cornec¹a*, Frédéric A. C. Le Moigne<fnref n="2" />,⁵b, Morgane Gallinari<fnref n="1" />, Brivaela Moriceau<fnref n="1" />, Jordan Toullec<fnref n="1" />, Morten Iversen<fnref n="3" />,<fnref n="4" />, Anja Engel<fnref n="5" />, Christina L. De La Rocha<fnref n="1" />.

<footnote n="1">¹ Univ Brest, CNRS, IRD, Ifremer, LEMAR, F-29280 Plouzane, France.</footnote>

<footnote n="2">² National Oceanography Centre, Southampton, European Way, SO14 3ZH, UK.</footnote>

<footnote n="3">³ Alfred Wegener Institute for Polar and Marine Research, Bremerhaven, Germany.</footnote>

<footnote n="4">⁴ MARUM and University of Bremen, Bremen, Germany.</footnote>

<footnote n="5">⁵ GEOMAR, Helmholtz Centre for Ocean Research Kiel, Düsternbrooker Weg 20, D-24105 Kiel, Germany.</footnote>

<footnote>a Now at Institute for Marine and Antarctic Studies, University of Tasmania 7001 AUSTRALIA.</footnote>

<footnote>b Now at Mediterranean Institute of Oceanography (MIO), UMR CNRS 7294 - IRD 235, UM AMU 110, 163 Avenue de Luminy, 13288 Marseille Cedex 09.</footnote>
Christina De La Rocha: xtinadlr@hushmail.com

Keywords : marine snow; Stokes’ Law; mineral ballast; sinking velocity; aggregation; zooplankton.
Abstract

Numerical simulations of ocean biogeochemical cycles need to adequately represent particle sinking velocities (SV). For decades, Stokes’ Law estimating particle SV from density and size has been widely used. But while Stokes’ Law holds for small, smooth and rigid spheres settling at low Reynolds number, it fails when applied to marine aggregates complex in shape, structure and composition.

Minerals and zooplankton can alter phytoplankton aggregates in ways that change their SV, potentially improving the applicability of Stokes' models. Using rolling cylinders, we experimentally produced diatom aggregates in the presence and absence of minerals and/or micro-zooplankton. Minerals and to a lesser extent micro-zooplankton decreased aggregate size and roughness and increased their sphericity and compactness. Stokes’ Law parameterized with a fractal porosity modelled adequately size-SV relationships for mineral-loaded aggregates.

Phytoplankton-only aggregates and those exposed to micro-zooplankton followed the general Navier-Stokes drag equation suggesting an indiscernible effect of micro-zooplankton and a drag coefficient too complex to be calculated with a Stokes' assumption. We compared our results with a larger dataset of ballasted and non-ballasted marine aggregates. This confirmed that the size-SV relationships for ballasted aggregates can be simulated by Stokes' models with an adequate fractal porosity parameterization. Given the importance of mineral ballasting in the ocean, our findings could ease biogeochemical model parameterization for a significant pool of particles in the ocean and especially in the mesopelagic zone where the particulate organic
matter:mineral ratio decreases. Our results also reinforce the importance of accounting for porosity as a decisive predictor of marine aggregate SV.
Introduction

The settling of marine particulate matter despite being complex in shape, structure and density has traditionally been modelled as simple spherical particles in order to ease the numerical computation of the many biogeochemical processes in which sinking particles are involved. Since the 70's (Lerman et al. 1974), Stokes' Law (Stokes 1851) has been used to represent particle sinking velocities (SV) in ocean models. Stokes' Law is shown in equation 1,

$$U_s = \frac{g(\rho_s - \rho_f)D^2}{18\mu} \qquad (1)$$

where $U_s$ is the SV of a sphere in m s⁻¹, g is the acceleration due to gravity (9.81 m s⁻²), $\rho_s$ and $\rho_f$ are the sphere and fluid densities in kg m⁻³, D is the sphere diameter in m and $\mu$ is the fluid dynamic viscosity in kg m⁻¹ s⁻¹. Stokes' Law is a solution of the low dimensionless Reynolds number (Reynolds 1883) limit of the Navier-Stokes equation. Initially, it was formulated for small perfect spheres settling in a quiescent fluid at low velocity, assuming that inertial forces are negligible compared to viscous forces. It was empirically found to be valid at low Reynolds number $Re < 0.5$,

$$Re = \frac{U_s D}{\nu} \qquad (2)$$

where $\nu$ is the seawater kinematic viscosity in m² s⁻¹. Stokes' Law conveniently allows estimation of the SV of a sphere based on density and diameter only, two particle properties readily measured using in situ imaging and chemical analyses. The SV of natural marine particles such as phytoplankton cells well characterized in term of geometry and composition can
potentially be modelled by Stokes’ Law (Waite et al. 1992) or empirically modified versions (Miklasz and Denny 2010) given small variations in cell shape and physiological state affecting its density (Smayda 1970). However, large marine aggregates (e.g. marine snow) are complex in shape and structure compared to the perfect spheres in Stokes' equation (Chase 1979). For natural large marine aggregates, the following general Navier-Stokes' drag equation (Eq. 3; Stokes, 1851) can be used to estimate SV,

$$U_{agg} = \left[ \frac{4 g(\Delta\rho)D}{3 C_D \rho_f} \right]^{1/2} \qquad (3)$$

where $U_{agg}$ is the aggregate SV in m s⁻¹, $\Delta\rho$ is the excess density in kg m⁻³ (= the difference between the aggregate and fluid densities; equivalent to $\rho_s - \rho_f$ in Eq. 1), and $C_D$ is the dimensionless drag coefficient. The drag coefficient depends upon the shape and surface structure (i.e. roughness) of the particle and is well defined for those with known shapes such as fecal pellets' (tabular, cylindrical, conical, etc.). As a consequence, the SV of fecal pellets has been successfully resolved (Komar et al. 1981; Ploug et al. 2008; Yoon et al. 2001), and does not present major difficulties for model parameterization (e.g. Jackson 2001). However, the complex shape, structure and roughness of phyto-aggregates complicate estimations of the drag coefficient and thus the SV (Alldredge and Gotschalk 1988; Johnson et al. 1996; Loth 2008). Particle SV parameterizations in numerical ocean models tend to avoid a strict size-dependency. Instead, they use theoretically and empirically derived equations (Maggi 2013). These equations account for a large number of SV drivers such as particle shape (and the non-negligible drag
associated), porosity, structure, fluid temperature and salinity (Jokulsdottir and Archer 2016; Kriest and Oschlies 2008). Complexity in simulation is necessary to mimic natural processes, but it also presents the huge inconvenience of assuming some of the many parameters that are poorly constrained by observation or experiments. Model results are then limited to a very specific experimental or biogeochemical context.

Density ρs and diameter D are the two most important particle characteristics accounted for in Stokes' Law calculations (Eq. 1). The latter is easily accessible by different imaging methods that can be used in situ or on experimental studies in the lab or at sea (e.g. Holographic Particle Velocimeter, Flow Cytometer And Microscope, Underwater Vision Profiler, Laser In Situ Scattering and Transmissometry, Laser Optical Particle Counter, Coulter Counter, light photography, etc.). In theory, aggregate density can be calculated from measurements of the relative mass fractions of the different solid components combined with measurements of particle dry weights and volumes to access the porosity (Eq. 1 & 3). In practice, individual and simultaneous measurements and calculations of both diameter and density on particles are rarely performed because of the difficulties they represent. Instead, these measurements are often made on a mixture of aggregates and rarely on individual particles due to the detection limits for very small amounts of solid components contained in a single particle. The fraction of aggregate volume occupied by seawater (i.e. the porosity) that also enters the calculations of aggregate density is inferred from measurements of aggregate volume or dry weight (Engel and Schartau 1999; Iversen and Ploug 2010; Laurenceau-Cornec et al. 2015b; Passow and De La Rocha 2006),
or from a calculation based on aggregate cell concentration and sizes (assuming limited variations of cell sizes) and aggregate volume (Moriceau et al. 2007). For similar particles (i.e. formed in the same conditions and from the same pool of POM), the relative fractions of the solid components within each particle are not subjected to large variations from one particle to another. The chemical composition of a mixture of aggregates is thus likely to reflect the overall composition of each individual aggregate. However, particle porosity varies with size as marine aggregates are polydisperse systems (i.e. composed of particles of non-uniform sizes) and have a fractal nature (Logan and Wilkinson 1990; Meakin 1988). In essence, this means that the rate at which the aggregate volume is filled with new particles during aggregation is lower than the rate at which the size increases. The direct consequences are that aggregate porosity increases with increasing particle size and thus large marine aggregates have a lower density than small aggregates of similar composition. Parameterization of particle density in Stokes' Law should thus account for these variations of particle density with size.

If Stokes' Law is to be valid for a certain type of marine particles, it is likely that these particles are roughly spherical (or 'round'), limited in size and compact (low porosity). Minerals are well known for their so-called 'ballast effect' which describes an increase of size-specific aggregate excess density and SV upon mineral incorporation (Armstrong et al. 2002; Klaas and Archer 2002). Additionally, aggregation in the presence of minerals promote the formation of more numerous, compact, spherical and smaller particles (Engel et al. 2009; Hamm 2002; Iversen and Ploug 2010; Jagt et al. 2018; Nowald et al. 2015; Passow and De La Rocha 2006; Passow et al.
2014; Schmidt et al. 2014). By promoting fragmentation processes, minerals limit the formation of large particles. This could be explained by a reduction of particle stickiness (i.e. the probability that two particles assemble upon collision) in the presence of minerals. In aggregates including organic and mineral matter, the POM acts as a glue which can lose its efficiency and lead to disaggregation when the POC:DW mass ratio falls below a value around 3-7% ‘called the POC carrying capacity’ which depends on the mineral nature (De La Rocha et al. 2008; Klaas and Archer 2002; Passow 2004). Other studies also suggested that repeated collisions occurring during coagulation processes can lead to more compact particles (Laurenceau-Cornec et al. 2015b; Logan and Wilkinson 1990). This mechanism could occur when aggregating particles have a low stickiness and need to collide repeatedly before adhesion occurs. Initially demonstrated for colloids, this mechanism called reaction-limited aggregation or RLA (Ball et al. 1987; Logan and Wilkinson 1990; Meakin 1988) could be extrapolated to larger particles. The opposite mechanism – the diffusion-limited aggregation or DLA (Logan and Wilkinson 1990; Meakin 1988; Witten and Sander 1981) – collate particles with a high stickiness and tend to form loose and porous aggregates.

To a lesser extent zooplankton can also influence particle sizes, shape, and structure.

Zooplankton feeding activity can fragment large aggregates into smaller particles, the so-called ‘sloppy feeding’ (Dilling and Alldredge 2000; Goldthwait et al. 2004). Zooplankton could also potentially alter the surface roughness and general structure of marine snow aggregates, but to our knowledge there are no reported observations of such an effect.
We tested the hypothesis that the presence of minerals and micro-zooplankton feeding can both (in interaction or separately) promote the formation of particles that Stokes’ Law describes well enough so their SV can reliably be predicted theoretically.

To test our hypothesis, we experimentally formed phytoplankton aggregates, in the presence and absence of minerals and micro-zooplankton to determine their effect on particle aggregation, and subsequent morphology and SV. Our observed size-SV relationships for the different treatments were then plotted against different parameterization of Stokes’ Law (Eq. 1) and the Navier-Stokes’ drag equation (Eq. 3).
Author Manuscript

# Material and methods

## Experimental setting

Two chain-forming diatom cultures — *Skeletonema marinoï* and *Chaetoceros gracilis* — were aggregated and kept in suspension in cylindrical rolling tanks (Shanks and Edmondson 1989) in the presence and absence of calcite (bio-mineral) and/or rotifers (micro-zooplankton) during 8 days in darkness and at 15°C. A total of twelve 4.5 L cylindrical tanks corresponding to three replicates of four different treatments (P: phytoplankton only; PZ: phytoplankton + micro-zooplankton; PM: phytoplankton + mineral; PMZ: phytoplankton + mineral + micro-zooplankton) were placed on a motorized roller table at t0 and rotated at a speed between 1.5 and 3 rpm. Variable rotating speeds were necessary to keep all particles formed in the different treatments in suspension and to account for SV changes along the experiment duration (see the Results section).

All twelve tanks were filled half with 0.6 µm filtered seawater and half with a mixture of *S. marinoï* and *C. gracilis* grown separately in a modified f/2 medium [Si(OH)₄ = 50 µM; PO₄ = 4 µM; NO₃ = 64 µM; vitamins and metals in f/2 quantities], and diluted to reach final tank concentrations of 65,000 and 110,000 cell L⁻¹ respectively. Calcite was prepared from Cretaceous chalk (see Le Moigne et al. 2013, for further details) and added to the tanks 3, 7 and 11 (PM treatment) and to the tanks 4, 8 and 12 (PMZ treatment), at a final concentration of 1.3 mg L⁻¹. This procedure led to a PIC:POC ratio of 0.1 at the beginning of the experiment, so comparable to but slightly lower than values generally observed for coccolithophore blooms as
stated in Le Moigne et al. (2013). Marine rotifers of the genus *Brachionus* sp. were added to the tanks 2, 6 and 10 (PZ treatment) and to the tanks 4, 8 and 12 (PMZ treatment), at an approximate concentration of 6200 individuals L⁻¹ (except tank 10 which received 4400 ind. L⁻¹). This concentration is within the range of typical rotifer concentration encountered in coastal water (up to 8000 ind. L⁻¹; Maioli Castilho and Arcifa, 2000).

Aggregate abundance, size and morphology

Variations in aggregation and fragmentation processes, abundance, size and morphological
properties of aggregates were monitored in the different treatments by imaging the particles
present in the tanks at each sampling time. On day 2, 4 and 8, four tanks corresponding to each
treatment were removed from the rotating table, placed on their side on a plexiglass plate and
aggregates were allowed a few minutes to settle to the bottom. Two cold optic fiber lights
(Schott KL 2500 LCD) illuminated the particles from two sides and a 10 megapixels image was
made from below through the bottom of the tank with a Canon Powershot G12 camera. All
images were analyzed using Fiji, a plugins-enriched distribution of the National Institutes of
Health open-source software Image J (Schindelin et al. 2012; Schneider et al. 2012). Pictures
were converted to binary (i.e. black and white) using the threshold function. All particles present
in the image were counted and analyzed in term of size and shape descriptors (e.g. circularity)
provided by Fiji. Following Laurenceau-Cornec et al. (2015a) particle size spectra were obtained
by converting particle projected areas to equivalent spherical diameter (ESD), and by binning all
Author Manuscript

particles into 14 logarithmically-spaced size classes (Table 1) to best characterize the whole size
range (Jackson 2005; Jackson et al. 1997). Bins containing 5 particles or fewer were excluded
from the analysis to ensure statistical relevance (Jackson et al. 2005). The lower size limit was
0.87 mm ESD and upper size limit was 129.81 mm ESD. Particle shape and structure were
described by their circularity and boundary fractal number D<sub>B</sub> taken as a metrics of surface
complexity (or roughness). D<sub>B</sub> also called box-count fractal dimension (or the ‘Kolmogorov
dimension’) is the slope of the regression line for the log-log plot of several sets of box sizes (or
grids) needed to cover the whole perimeter of the particle and the number of pixels counted
inside the boxes for each set of grids (for further explanation on the box-counting method see
Kaye et al. 1987; Russ, 2011). D<sub>B</sub> was obtained from a box-counting fractal analysis using the
Image J plugin FracLac (Karperien 2005). For this fractal analysis a special subset of images
consisting of vignettes of a few aggregates that shared the same and best light conditions (i.e. at
the intersection of the two light sources) was extracted from the original image. These vignettes
were thresholded separately to optimize particle boundary detection and reassembled in collages
of aggregates thus offering the best details of their whole perimeter. In addition, only the
particles larger than 0.78 mm<sup>2</sup> in area (equivalent to 1 mm in ESD) were included to give the
best results based on the image native resolution. It must be noted that in other sections of the
image analysis (e.g. particle size, number and shape), this level of details on particle surface was
not required and initial images displaying all particles were used. Bare outline masks images
from the Fiji particle analyses were scanned with FracLac by selecting 12 grids per particle (so
12 scans per particle; each scan having a different box size) and letting all other parameters such as maximum and minimum box sizes automatically set by the software.

Aggregate solid content

In order to infer aggregate densities required for the theoretical SV calculations (using Eq. 1 and 3), the composition of the whole aggregated matter was measured in each tank. After the images were taken, all aggregated matter that could easily be identified (i.e. with a diameter of roughly 1 mm at the minimum) was picked inside each tank and combined in a same beaker using a 10 mL disposable pipette on which the tip had been cut off. For each individual aggregate sampled particular care was taken to avoid any loss of material in the process which would have biased the analyses of the relative fractions of their solid constituents. The aggregate mixture was then subsampled (1-3 replicates of 5 to 15 mL) and analyzed for Particulate Organic Carbon (POC) and Nitrogen (PON), biogenic silica (bSiO₂), Particulate Inorganic Carbon (PIC) and Transparent Exopolymer Particles (TEP). TEP being measured following a semi-quantitative method were not used for density calculations but only as a qualitative parameter. Tank water that was cleared from all visible aggregates — hereafter called ‘background water’— was also sampled (1-3 replicates of 10-250 mL) and analyzed to estimate its content in POC, PON, bSiO₂, PIC and TEP. The exact particulate matter concentrations in aggregates were obtained by measuring the particulate content in the small volume of water inevitably sucked into the pipette during aggregate sampling and subtracting it from the aggregate sample concentrations. All
protocols followed for POC, PON, bSiO₂ and PIC sample preparation and analysis are described
in Le Moigne et al. (2013). TEP were measured using the spectrophotometric method from
Passow and Alldredge (1995). A 0.02 % aqueous solution of Alcian Blue in 0.06 % acetic acid
was used to stain the TEP contained in samples. In each tank three replicates of 10 mL of
background water and 10 mL of the aggregate mixture were sampled and filtered onto 0.4 µm
Whatman polycarbonate filters at a constant vacuum below 200 mbar to prevent the TEP from
being pulled through the filter pores. The filters were then stained with 500 µL of 0.02 % Alcian
blue solution and soaked for 2h in 6 mL of 80 % H₂SO₄ to dissolve the stain. The absorbance of
the solution was measured at 787 nm with a PRIM Light and Advanced Spectrophotometer. The
calibration curve relating mass of xanthan gum to absorbance was established using a protocol
modified from Passow and Alldredge (1995) to avoid problems potentially linked to a
heterogeneous xanthan gum solution. Instead, 0.4 µm polycarbonate filters were disposed onto a
Mettler Toledo scale with a sensitivity of 1 µg, tared, and 40 to 800 µg of xanthan gum powder
(VWR, USP/NF) were deposed directly onto the filters. These filters were then stained with 500
µL of 0.02 % Alcian blue solution at a constant vacuum under 200 mbar and soaked for 2 h in 6
mL of 80 % H₂SO₄. Finally, absorbance of the solution was measured at 787 nm.

Aggregate volume, porosity and density

Aggregate density (ρ<sub>agg.</sub>) was calculated in each tank from the volume fractions in the solid
content of POM (taken as the sum of POC and PON), PIC and bSiO<sub>2</sub> in the aggregate mixture —
used to calculate the density of the aggregate solid content ($\rho_{sol.}$) — and from the aggregate porosity (i.e. fraction of the aggregate volume occupied by water) using the following formula:

$$\rho_{agg.} = \rho_{sol.} \times (1 - P) + \rho_{sw} \times P \quad (4)$$

where $\rho_{sw}$ is the density (kg m⁻³) of the seawater contained in the aggregate and $P$ is the aggregate porosity. Densities for POM, PIC and bSiO₂ were respectively set at 1060, 2710 and 2000 kg m⁻³. Porosity was calculated using,

$$P = 1 - \left( \frac{Vol_{POM+PIC+bSiO_2}}{Vol_{agg.}} \right) \quad (5)$$

where Vol$_{POM+PIC+bSiO_2}$ is the total volume of the solid content in aggregates and Vol$_{agg.}$ is the total volume of aggregates estimated from image analysis. Only aggregates larger than 1 mm ESD were included to keep only those that were manually picked for the chemical analyses. Individual aggregate volume was calculated from the minor (min. ax.) and major axes (maj. ax.) of the best fitting ellipse on particle images given by Fiji and assuming a prolate shape, using the following formula:

$$Vol_{agg.} = \frac{4}{3} \times \pi \times \left( \frac{\min. ax.}{2} \right)^2 \times \frac{maj. ax.}{2} \quad (6)$$

Aggregate excess densities ($\Delta\rho$) were calculated assuming a seawater density of 1025.3 kg m⁻³ (corresponding to 15 °C and a salinity of 34).

## Sinking velocity measurements

### *Observations*
Aggregate SV were measured in vitro over the course of the experiment using the non-invasive video recording technique developed by Ploug et al. (2010). This method permits continuous measurements during incubation. It allowed to monitor the SV in each treatment at a higher temporal resolution (day 2, 3, 4, 5, and 8; Table S1, supplementary material) than other measurements requiring to end the incubation. Calculations are based on incremental x- and y-positions of particle orbital trajectories in relation to the center of the tank side being taken as a spatial reference. Measurements aimed to cover the largest size range offered by the particle population inside the tank but particle choice was also constrained by their adequate position and orbits required for post-recording analyses. A cold optic fiber light from a Schott KL 2500 LCD was used to illuminate the rotating particles without disturbing their orbital trajectories and video recording was conducted with the same Canon Powershot G12 camera used for particle imaging. The method from Ploug et al. (2010) was completed here by the use of a laser-pointer to identify in the video sequences each individual aggregate being filmed and record its distance from the side wall of the tank along an axis parallel to the rotation axis (z-axis). In the video footage, the apparent x, y coordinates and size of the aggregates are affected by its distance from the side wall of the cylinder through which the video is shot due to the deformation of perspective. Each x and y-positions of the aggregates and their size were corrected from this distortion effect using their distances from the side wall recorded for each particle. A total of 203 paired measurements of aggregate size and SV was conducted.
## Theory and models

Theoretical aggregate SV were obtained using Stokes' Law (Eq. 1) and the Navier-Stokes equation which includes the drag coefficient in its formulation (Eq. 3). Aggregate excess densities ($\Delta\rho$) required in both equations were calculated using two different parameterizations for aggregate porosities and introduced in each formulation, leading to four different models that were compared to observations.

### Model 1: Stokes' Law (Eq. 1) with a constant aggregate density

Here, a constant density was assumed for all aggregates having the same origin (from the same treatment and same tank). Aggregate densities were inferred from the relative fractions of their solid components in POM, bSiO₂, PIC measured on the aggregate mixture (see section 'Aggregate volume, porosity and density'). Porosities were fixed at 0 and 99% to mimic solid spheres and highly porous particles close to values reported in the literature for large marine aggregates such as marine snow (Alldredge and Gotschalk 1988).

### Model 2: Navier-Stokes drag equation (Eq. 3) with a constant aggregate density

In model 2, aggregate densities were calculated from the measured solid content (similar to model 1) and a constant porosity of 99%. Empirical drag coefficients were fixed at values of 1 and 5.

### Model 3: Stokes' Law (Eq. 1) with aggregate density scaled on a fractal geometry
Model 3 used a constant density for aggregate solid contents from measurements but assumed a porosity increasing with increasing size based on a fractal scaling using the formula proposed by Logan and Wilkinson (1990) and following Laurenceau-Cornec et al. (2015b):

$$P = 1 - a \times l^{D_3 - 3} \qquad (7)$$

where *P* is aggregate porosity, *l* is the largest diameter, *a* is an empirical constant and *D*<sub>3</sub> is the three-dimensional fractal number. Figure 1a shows the size-porosity relationship for different values of *a* and *D*<sub>3</sub>. Note the constant porosities (horizontal lines) at all sizes obtained for *D*<sub>3</sub> = 3 that corresponds to a Euclidean geometry for which particle volume is filled at the same rate than particle size increases. In this case *P* = 1 - *a*, and the porosity depends only on the value attributed to *a*. For small values of *a* and large *D*<sub>3</sub>, the rate at which porosity increases with size is maximum at the smallest sizes. For model 3, *D*<sub>3</sub> was fixed at 1.4 and 1.8 following values obtained in the literature and describing a large range of marine aggregate structures (Logan and Kilps 1995). The constant *a* was fixed at 0.03 following Logan and Wilkinson (1990).

## Model 4: Navier-Stokes drag equation (Eq. 3) with aggregate density scaled on a fractal geometry

Model 4 infers both the porosity and drag coefficient assuming a dependence on size based on a fractal scaling. Variation of the drag coefficient with size for various values of *D*<sub>3</sub> and *a* is shown in Figure 1b. The fractal porosity was calculated as described above with *D*<sub>3</sub> values of 1.4 and 1.8 and *a* = 0.03. Calculations of the fractal drag coefficient followed Laurenceau-Cornec et al. (2015b). Eq. 3 was solved for the drag coefficient using two paired values of size and SV
defining specific drag conditions [called ‘specific conditions’ in Laurenceau-Cornec et al. (2015b)]. A bisection algorithm (cost function; code available in supplementary material) computed on Matlab (Simulink) allowed to solve the White function numerically (White 1974) and to calculate the drag coefficients at each size from Reynolds numbers using Stokes' assumption as a starting point ($Re_0 = 24 / C_{D0}$). The White function is one of the numerous different empirical formulations for the drag coefficient (Clift and Gauvin 1971; Kelbaliyev 2011) and was chosen here for its simple expression. Aggregate solid fractions needed in the calculations of Figure 1b were arbitrarily set at 60% POM and 40% bSiO$_2$; these values correspond roughly to the measurements obtained for the P treatment on day 9 (see Results section). For large values of $a$, Stokes’ Law and the White function give similar $C_D$ values up to very large sizes (e.g. specific drag condition 1, $a = 0.1$) suggesting that Stokes’ Law could be valid in these conditions. Note the poor control of $D_3$ on $C_D$ variations except at very small sizes and coefficient $a$.

## Error assessment and statistical analysis

The quadratic propagation of error from the potential erosion or dilatation of particle projected areas during image thresholding (see section ‘Aggregate abundance, size and morphology’) to particle porosity and density calculations was assessed using rules derived from the Gaussian equation for normally-distributed errors. Particles that presented most of the alteration after threshold were the largest and most complexly-shaped. For these particles a homogeneous
illumination of the whole surface was difficult. A filter fixed at 3 mm² projected area (equivalent to a 2 mm ESD) and 0.5 circularity was applied to each tank binary image to keep only the particles suspected to have generated most of the error. Then the effect of eroding or dilating these particles on the total aggregate volume was assessed for each tank. Our results show that a removal of a layer of 1 pixel from these particle contours (erosion) induces an average total aggregate volume loss of 7.2 ± 3.8 %, while adding a layer of 1 pixel (dilatation) induces an average total volume gain of 10 ± 4.5 %. Based on this, calculations of propagated errors on porosity and density assumed a ± 10 % variation of particle volume due to image binary conversion. Standard deviations of the replicate measurements of POC, PON, bSiO2 and PIC contents in aggregates were also used in the calculations of error propagations.

One-way analyses of variance (ANOVA) were conducted using the language and environment for statistical computing R to test the statistical significance of the differences in the mean ESD, particle boundary fractal numbers and circularities between treatments. Homogeneity of variances were tested using a Bartlett's test, and non-parametric Kruskal-Wallis rank sum tests were also used as means to confirm treatment differences. All pairwise comparisons between treatments were performed following the Tukey's method (differences in mean levels) to identify difference between treatments.

Non-linear (power) regression fits to the size-SV relationships for this study and others shown here were also performed with R. Along with the regression coefficient, standard errors, *p*-values
and profile likelihood confidence intervals (95 %) were derived from the model fits and are provided as possible metrics to evaluate the goodness of fits.

To assess the performance of the four model parameterizations to predict experimental data, the Residual Sum of Squares (RSS) and Residual Mean Square Error (RMSE) between predicted and observed values were calculated. Smallest values of RSS and RMSE were considered to indicate the best model fit to observations. Additionally, the difference in logarithms of the likelihood Δlog(L), based on a Gaussian error distribution with a constant variance of log-transformed data, was taken as a criterion to inter-compare model performance (Armstrong et al. 2002, and references therein). The log(L) was calculated as in Moriceau et al. (2009):

$$ \log(L) = -\frac{N}{2} \log \left( \frac{\sum (\log(\hat{SV}_i) - \log(SV_i))}{N} \right) \quad (8) $$

where N is the number of data points, SVᵢ is the measured SV at data point i, and ̂SVᵢ is the corresponding value predicted by the model. After pairwise model comparisons, a difference in log(L) of at least two points was considered as a criterion of a better fit to the observations (Hilborn and Mangel 1997).
# Results

## Aggregation-disaggregation processes

The presence or absence of calcite and micro-zooplankton induced significant variations in particle size, shape and general aspect (Fig. 2). Direct observations made on day 4 (Fig. 2a to d) suggest that the addition of calcite and micro-zooplankton (PZ, PM and PMZ) promoted the formation of smaller aggregates than with phytoplankton alone (P). Aggregates in the P and PZ treatments appeared more ‘fluffy’ (i.e. porous) and less round than in the PM and PMZ treatment, suggesting that calcite could induce an increase of particle compactness and sphericity, an effect not observed in the presence of micro-zooplankton alone. By day 8 (Fig. 2e to h), most of the differences between treatments previously visible by direct observations diminished significantly. Small tails appeared at particle surfaces in the PM and PMZ treatments (Fig. 2g and h, white arrows) as it is often observed on fast-sinking particles (see for example Fig. 1 in Engel et al. 2009).

Aggregates formed between days 0 and 4 in the following chronological order: PM and PMZ first, then PZ and finally P being the last to form. The maximum number and volume of > 1 mm-sized aggregates were reached on day 4 in the P and PZ treatments, while it peaked on day 2 in PM and PMZ treatments (Table 2) suggesting that faster aggregation occurred in the presence of calcite. After the maxima in aggregate number and volume a decreasing trend was observed in all treatments. Monitoring of particle size spectra (Fig. 3) confirmed that most of the aggregation processes took place in all treatments from day 2 to 4 (no data for the P treatment for which no

Author Manuscript
visible aggregates were formed on day 2). This result was revealed by a transfer of particles from small size classes to larger ones. In the PZ, PM and PMZ treatments, the number of aggregates in size classes ranging from 1 to 3 mm ESD approximately varied the least during aggregation processes, suggesting that this particular size range represented the most stable size reached in these treatments. An increase of the smallest particles in PM tanks between days 2 and 4 showed that disaggregation processes also took place in this treatment during this time period. Days 4 to 8 were mostly marked by disaggregation processes illustrated by less of the large particles combined with more of the small ones. Particles sized around 1-2 mm were again those that participated the least to disaggregation processes in the PM and PMZ treatments confirming the equilibrium status of this size range. Furthermore, in the PM treatment the size class 8 (1.1 - 1.5 mm) was the only class that showed an increase in particle number, while both smaller and larger particles displayed losses. The size class of particle that varied the least between days 4 and 8 in both P and PZ treatments were smaller around 0.4 - 0.5 mm.

Aggregate characteristics

Aggregate size, circularity and boundary fractal number

During the experiment the largest mean ESDs (and also largest standard deviation) was observed on day 4 in the P treatment with a value of 2.53 ± 1.28 mm (Fig. 4a; Table 3), suggesting that these aggregates were the largest but also the least uniform in size. On day 8, aggregates from the P treatment had the smallest mean and median ESDs. All other aggregates (PZ, PM and PMZ
treatments) had more uniform sizes and displayed little variations during the experiment, confirming results from size spectra variations (see above). On day 4, statistical tests (ANOVAs, Kruskal-Wallis test and Tukey pairwise multiple comparison of means) confirmed a significant difference between the mean ESDs of P aggregates and those from other treatments except with PMZ (Table 3; Tukey *p*-value = 0.217 for the pair PMZ-P). No statistical difference could be found between the mean ESDs of all treatments on day 8, confirming direct observations that aggregates became more similar toward the end of the experiment.

Analyses of aggregate circularities showed that particles formed in the presence of calcite and micro-zooplankton (PZ, PM, PMZ) were the most circular (Fig. 4b and Table 3). Conversely, P aggregates had the lowest median circularities, agreeing with results suggesting more complex shapes for these aggregates (see above). ANOVAs and Kruskal-Wallis tests confirmed the statistical differences of circularity between treatments (Table 3). Pairwise multiple comparison of means (Tukey contrasts) showed that P aggregates were significantly different in circularity from other treatments except on day 8 for which no statistical difference could be made between P and PM aggregates.

Aggregate boundary fractal number from image analyses mirrored the circularity results and suggested that aggregates formed in the presence of calcite and micro-zooplankton had less complex surfaces (i.e. were smoother) than if produced from phytoplankton alone (Fig. 4c). For each sampling day, box-counting analyses showed that aggregates formed in the presence of calcite (PM and PMZ treatments) had the lowest median boundary fractal number *D*<sub>B</sub> (Fig. 4c
and Table 3), except on day 8 for which the PZ treatment showed the lowest mean $D_B$. ANOVAs and Kruskal-Wallis tests confirmed the statistical significance of the differences in mean $D_B$ between treatments (Table 3). Pairwise multiple comparisons of means (Tukey contrasts) showed in particular that P treatments significantly differed from others except for the pairs PZ-P and PM-P for which no statistical difference could be established on day 4 and 8 respectively (Fig. 4c and Table 3). The largest variation of median $D_B$ was observed for the PZ treatment between days 4 and 8 with a 0.034 decrease suggesting a significant surface smoothing of the particles.

Aggregate composition and porosity

An increase of all total particulate matter (POM, PIC and bSiO₂) content was observed between days 2 and 4 in all treatments— so during on-going aggregation processes— while a decrease of POM concentration occurred between days 4 and 8 suggesting degradation processes (Table 2). As expected, relative mass fractions of POM, PIC and bSiO₂ in aggregate solid contents (see Table S2 in supplementary material for volume fractions) showed large differences between treatments, especially those that were grown in the presence of calcite or not. Dry content of aggregates from the PM and PMZ treatments contained in average 54.8 ± 8.4 and 51.2 ± 7.9 % of PIC respectively, while those from the P and PZ treatments had 4.4 ± 2.4 and 4.1 ± 1 % of PIC, confirming that calcite was important for aggregation in the PM and PMZ treatments. POM represented the largest fraction of the solid content in P and PZ treatments (average values 69.9 ± 15.9 % and 69.7 ± 7.1 % POM respectively). Decreases of the fraction of organic components
(POM) between days 4 and 8 were accompanied by large increases of bSiO₂ toward the end of the experiment (increases of 25.9 ± 2.3, 4.2 ± 1.7, 4.1 ± 1.3 and 6.2 ± 0.8 % bSiO₂ in P, PZ, PM and PMZ treatments respectively). Between days 2 and 8, the PIC fraction decreased continuously in the P and PZ treatments (decreases of 3.4 ± 0.9 and 1.9 ± 0.4 respectively) while it increased in the PM and PMZ treatments (increases of 16.4 ± 10.9 and 14.1 ± 2.5 respectively). Total TEP content in aggregates increased between days 2 and 4, and decreased toward day 8 in all treatments, except in the P treatment where TEP increased between days 4 and 8 (Table 2). Aggregates from the P treatments had the highest porosities (average of 99.76 ± 0.1 % of seawater). The lowest porosities were observed for the PM and PMZ treatments on day 8 (99.33 ± 0.11 and 99.32 ± 0.11 % of seawater respectively), suggesting that these particles had the most compact structure.

As expected, aggregates that incorporated calcite had the highest excess densities illustrating the ballast effect (averages of 3.73 ± 1.7 and 3.57 ± 1.5 kg m⁻³ in the PM and PMZ treatments for the whole duration of the experiment), while aggregates from the P and PZ treatments had the lowest excess densities (averages of 0.6 ± 0.42 and 1.27 ± 0.38 kg m⁻³ in the P and PZ treatments respectively).

Observed sinking velocities and model predictions

Measured SV presented very different ranges between treatments that contained calcite and those that did not (Fig. 5 and Table 4). Ballasted aggregates (PM and PMZ treatments grouped) had an
average SV of 1045 ± 310 m d⁻¹ (min = 327 m d⁻¹; max = 1704 m d⁻¹) while aggregates without added calcite (P and PZ treatments grouped) had an average SV of 456 ± 136 m d⁻¹ (min = 218 m d⁻¹; max = 928 m d⁻¹). This result confirms the expected ballast effect induced by the addition of calcite. The goodness of the fits to the relationship between ESD and SV varied as a function of sampling day and treatment. Overall, power law regressions presented a better fit to the relationship between ESD and SV for aggregates formed in the PM and PMZ treatments with an improvement over time towards the end of the experiment (Table 4; lowest standard errors and *p*-values < 0.001). Conversely, higher standard errors and *p*-values on coefficient estimates were obtained when fitting power law regressions to the data of the P and PZ treatments. This result suggests that size is a better predictor of the SV for ballasted aggregates than for those grown with phytoplankton only. Conversely, the goodness of the fit to the size-SV relationship was not improved by the addition of zooplankton (comparisons P vs PZ and PM vs PMZ), suggesting that, unlike calcite, zooplankton do not increase the control of size on aggregate sinking velocities.

Fits of the four models using Stokes' Law and the Navier-Stokes drag equation (see section 'Theory and models') to the observations are reported in Figure 5. Quantitative estimation of model performances using the Residual Sum of Squares (RSS) and the Root Mean Square Errors (RMSE) showed that model 4 (Navier-Stokes drag equation parameterized with an aggregate density scaled to fractal geometry; D₃ = 1.4; *a* = 0.03; Fig. 5d) presented the best fit to the P and PZ data (Table5; average RSS of 0.16 ± 0.15 and average RMSE of 0.1 ± 0.03). Also, model 3
(Stokes' Law parameterized with an aggregate density scaled to fractal geometry; D₃ = 1.8; a = 0.03; Fig. 5c) best reproduced PM and PMZ data (Table 5; average RSS of 0.43 ± 0.51 and average RMSE of 0.11 ± 0.04). The criterion of a minimum two-points higher difference in log(L) between models 2 and 4 for the P and PZ treatments confirmed better performance for model 4 in tanks 9 and 2 [Δlog(L) of 4.29 and 2.18 respectively]. In the PM and PMZ treatments, model 3 presented a higher log(L) of at least 2 points than model 1 in tanks 3, 7 and 4 [Δlog(L) of 2.97, 8.9 and 2.75 respectively], while model 1 performed better than model 3 only in tank 11 [Δlog(L) of 3.92] where the greatest SV were observed confirming the RSS and RMSE results.
# Discussion

## Uncertainties from sampling and temporal resolution

Aggregates that were sampled and grouped in a mixture at the end of each tank incubation had a minimum diameter of ca. 1 mm (see section ‘Aggregate solid content’). Fractions of POM, bSiO₂ and PIC were measured in these mixtures and used to calculate general densities for the aggregated material formed in each tank. Matter smaller than 1 mm was considered unidentified and intentionally excluded from our sampling to avoid any possible contamination of our mixtures. Despite this precaution, uncertainty lies in potential variations of the solid component fractions across the population of individual aggregates that composed each mixture. This risk is high for aggregates formed from natural phytoplankton communities (Alldredge and Silver 1988; Laurenceau-Cornec et al. 2015b) but we believe that our aggregates formed in controlled conditions from uniform phytoplankton cultures presented very similar composition. The limited scattering of the data points originating from similar treatments (Fig. 5) combined with the ability of our models using these measured compositions to reproduce fairly well the observations gives us good confidence that these variations were limited.

Another uncertainty pertains to the temporal resolution of our sampling. Because of logistic constrains related to our numerous measurements, sampling times were restricted to day 2, 4, and 8, with two additional times at day 3 and 5 for the SV measurements only. It implies that fluctuations in aggregate physico-chemical properties during ongoing aggregation/disaggregation or remineralisation could have occurred between two consecutive sampling points without being
resolved by our measurements. Our results mainly reflect the general trends between two time
steps and can identify the transient states and especially the tipping points at an accuracy scaling
between one and four days. For instance, the onset of degradation as possibly suggested by a
transition from TEP increase to decrease at day 4 (Table 2; exception for P treatment) might have
occurred earlier (between day 2 and 4) or later (between day 4 and 8) and at a different peak
value. Results and conclusions drawn here are however not highly sensitive to this uncertainty
since we mostly relate measurements of aggregate physico-chemical properties to their sinking
velocity measured at near or similar sampling times with little focus on the effect of transient
states.

Minerals and zooplankton roles in Stokes' Law applicability

Our initial hypothesis was that incorporation of minerals and zooplankton feeding on marine
aggregates induce modifications of particle properties that allow a reliable use of Stokes’ Law.
Results from our experiment confirmed that particles formed in the presence of calcite (PM and
PMZ treatments) have a size-SV relationship that consistently follows Stokes’ models. In
particular, the overall best fit to the PM and PMZ size-SV data was obtained for model 3
(parameterization of Stokes’ Law with a porosity scaled to a fractal geometry, Fig. 5c).

Interestingly, model 1 (parameterization of Stokes’ Law with a constant porosity) showed better
prediction than model 3 for the fastest-sinking aggregates that were formed at the end of the
incubation (tanks 11 and 12 on day 8). This result suggests that aggregates from tanks 11 and 12
were the best candidates to fit a traditional Stokes' Law model. These aggregates would have acquired properties that, among all particles formed in this experiment, were the closest to those of a 'Stokes' particle' (i.e. a small perfectly smooth sphere settling at low Reynolds number). Average ESDs of aggregates from tanks 11 and 12 were the largest among their respective treatments (PM and PMZ), thus not supporting a small size as a possible explanation for their better fit to a traditional Stokes' curve. However, their low porosities (Table 2) suggest that these particles were very compact and likely had a structure better represented by a Euclidean geometry than a fractal geometry. This observation tends to confirm the suspected effect of minerals on decreasing particle stickiness and enhancing aggregate compactness through Reaction-Limited aggregation. The fractal nature of marine aggregates implies that small ones are expected to be more compact than large ones due to the increase of porosity with size as already observed experimentally by Engel et al. (2009). By leading to a loss in stickiness, the presence of minerals could counteract to some extent the fractal building of marine aggregates by forcing each building block to more efficiently fill the volume available and follow to a higher extent a Euclidean geometry. One would then expect mineral-loaded aggregates to have a *D*<sub>3</sub> fractal number higher than those usually observed for fractal marine aggregates. Logan and Wilkinson (1990) compared the fractal numbers of various marine aggregates formed in different environments. They reported fractal numbers for marine snow of 1.26 (Alldredge and Gotschalk 1988) while aggregates collected in an estuary – thus having a higher terrigenous mineral fraction – had a higher fractal number of 1.78 (Gibbs 1985). Boundary fractal numbers *D*<sub>B</sub>
reported here are however not comparable with values of one-, two-, or three-dimensional fractal
numbers (noted D₁, D₂, and D₃ respectively and themselves inter-comparable only in specific
conditions) as they are similar in meaning but use different calculation methods (Kilps et al.
1994; Russ 2011). However, low values of D₃ as observed especially for tanks 11 and 12 on day
8 tend to indicate a low surface complexity as expected for more compact and smooth particles.
Particles formed in the presence of micro-zooplankton, but without calcite (treatment PZ), did
not show marked differences from the P treatment, suggesting a limited effect from micro-
zooplankton presence alone. This result does not confirm our initial hypothesis that micro-
zooplankton could alter the size-SV relationship in a similar manner to calcite. PZ aggregates on
days 2 and 4 showed significantly smaller circularities and higher D₃ values than for the PM and
PMZ treatments but had on day 8 all properties allowing to expect a much better fit to a Stokes’
Law model. Only excess density was significantly larger for the calcite treatment compared to
the micro-zooplankton treatment on day 8 due to the ballasting effect of calcite. This result is
nevertheless counter-intuitive as Stokes’ Law is deemed valid at Re < 0.5. This means that at
constant seawater kinematic viscosity ν (see Eq. 2), Stokes’ Law should be valid for small and
slowly-sinking particles. Regardless of the treatment, none of the particles formed in this study
had a Re value lower than 0.5 (range between 2 and 50; Table 4), and our results suggest that the
fastest-sinking particles, those ballasted with calcite, were those that agreed best with Stokes'
models. Several previous studies of natural and artificially-formed marine snow reported values
of Reynolds number comparable to ours (Alldredge and Gotschalk 1988; Engel et al. 2009). This
is precisely the range in which $C_D$ starts to differ significantly between Stokes’ Law and other empirical approximations like Whites’ (see for example Fig. 1 in Loth 2008). The $Re < 0.5$ limit for the validity of Stokes’ Law is empirical and based on fluid flow observations past spherical sinking particles (Loth 2008; White 1974). Our results suggest that for $1 < Re < 50$, Stokes’ regime may remain reasonably applicable for phytoplankton aggregates given that they fulfil certain criteria in terms of roundness, roughness and compactness that could possibly be acquired in case of mineral incorporation. This result provides insights as to why the PZ aggregates — satisfying these criteria and having Reynolds numbers lower than those observed for the particles that agreed best to the Stokes’ model (tanks 11 and 12) —, did not fit better to Stokes’ models. Different drag properties that were not adequately captured by our boundary fractal number analysis could explain this discrepancy. Maggi (2015) showed experimentally that internal and external fractal characteristics of the aggregates can induce non-linear hydrodynamic effects through their influence on surface roughness and porosity. In particular, feeding activity of the rotifers at the surface of the aggregates could have led to an increase of surface roughness at a scale unexpectedly not resolvable on our images given rotifers’ small size (usually a few hundred microns) compared to our millimeter-sized aggregates. In addition, rotifers used in this experiment are micro-zooplankton and cannot serve as a model for all zooplankton. It is likely that larger species would have interacted differently with aggregates and probably led to contrasting results.
Conversely, the best fits for all non-calcite ballasted particles (P and PZ treatments) were obtained with the Navier-Stokes drag equation. Model 4 parameterized with a fractal porosity and drag performed well. This result confirmed our expectations that aggregates grown in absence of calcite should have characteristics farther from a theoretical Stokes' particle because of their more irregular shape, higher porosity and more complex fractal structure. It is also in agreement with the numerous studies that documented the poor control of size on SV for large, loose and highly-porous phytoplankton aggregates. This can be partly explained by the increase of porosity with size for fractal aggregates that can even lead to a decrease of the SV with size for the largest particles (Laurenceau-Cornec et al. 2015b).

## Relevance to other ballasted and non-ballasted aggregates

To test whether our findings remained valid with a larger dataset, we gathered size-SV data from other numerous experimental and field studies (Alldredge and Gotschalk 1988; Alldredge and Gotschalk 1989; Azetsu-Scott and Johnson 1992; Carder et al. 1982; Diercks and Asper 1997; Engel et al. 2009; Gibbs 1985; Iversen et al. 2010; Iversen and Ploug 2010; Iversen and Robert 2015; Jouandet et al. 2011; Kajihara 1971; Laurenceau-Cornec et al. 2015b; Nowald et al. 2009; Ploug et al. 1999; Shanks and Trent 1980; Syvitski et al. 1995). We refer the reader to Laurenceau-Cornec et al. (2015b) if more information on these studies (particle type, method used, location) are needed. We split data into two categories – ballasted and non-ballasted aggregates – and fitted on these two groups parameterizations of models 3 and 4 (Fig. 6). It is
important to note that this classification is arbitrary and that drawing a clear limit between these
two types of aggregates cannot represent the reality as it exists a continuum of mineral loads in
aggregates — from none (i.e. phytoplankton that do not produce mineralized skeletons) to those
that include minerals produced naturally by phytoplankton, and exogenous minerals scavenged
in the water column. Empirically, we considered all aggregates without exogenous mineral as
non-ballasted (note that our P and PZ treatments contained biogenic silica from the diatom
frustules) and all aggregates that included exogenous minerals as ballasted (one exception noted
on the figure caption). For non-ballasted aggregates, Figure 6a shows that the use of model 4
(Navier-Stokes drag equation) parameterized with various fractal numbers, initial conditions and
coefficient *a* fitted the measured size-specific SV best. Again, this confirms the known
complexity of predicting the SV of phytoplankton aggregates from their size and density alone. It
also highlights the crucial importance of the fractal structure and drag characteristics in the
control of the SV for these aggregates and that no reliable prediction of the SV could be achieved
without considering these particle characteristics. Figure 6b shows various datasets of aggregates
that incorporated minerals either in situ or artificially in experimental conditions. With the
exception of a set of aggregates, most of the size-SV relationships display a similar slope. Our
model 3, a Stokes' Law model parameterized with a fractal porosity, reproduces well this general
slope with a fractal number of approximately 1.8. The different SV offsets needed to fit the
model to each dataset are easily obtained by varying the coefficient *a* between 0.1 and 0.001.
This result strongly supports our findings and confirms that a Stokes' parameterization can
reliably be used to predict the SV of aggregates including minerals. It also highlights the already
noted importance of the dimensionless empirical constant *a* in this parameterization
(Laurenceau-Cornec et al. 2015b) and poses the question of how to constrain it. Equation 6
suggests that at a given *D*3 and particle size, the porosity is directly proportional to *a*. This
indicates that assuming a known *D*3 — set for instance at 1.8 based on Figure 6b — a prediction
of the SV of ballasted aggregates as a function of their size and density would be possible using
Stokes' Law but only given data on the value of *a*. At similar aggregate sizes and densities, the
SV offset between the various sets of aggregates would thus be created only by slight differences
in porosity, with the highest SV corresponding to the lowest porosities and vice versa.

We investigated the extent to which the concentration of minerals in the aggregates could be
important in determining the acquisition of Stokes' like properties by the particles. In all studies
reported in Figure 6b, aggregates carried between ~7 and 98% of various minerals (mineral:dry
weight ratio), which is a very large range of mineral load. This gives us confidence that the
mechanism identified here could be highly relevant to all mineral-loaded aggregates possibly
regardless of mineral nature. According to Le Moigne et al. (2014), surface POC flux associated
with minerals could represent 60 % of the POC flux in the Atlantic and 40 % in the Southern
Ocean and Le Moigne et al. (2012) estimated over 55 locations in the North Atlantic and
Southern Ocean that approximately 32 % of the sinking POC flux was associated with minerals.
Atmospheric dust deposition that brings lithogenic particles to the surface ocean are important
events that increase the proportion of mineral-associated POC in the surface ocean (Bressac et al.
2014; Guieu et al. 2014; Pabortsava et al. 2017). Interestingly, the increase of the POC:mineral ratio in aggregates in relation to ocean acidification has been shown to significantly alter the size-SV relationship with clear deviations from Stokes' Law (Biermann and Engel 2010). It is also important to consider how the proportion of POC:mineral changes with depth. The preferential remineralization of POC through the upper mesopelagic results in an increase of the mineral fraction inside the aggregates (Honjo et al. 1982; Iversen and Ploug 2010). Mineral dissolution which also takes place with depth has been shown to occur at limited rates in large marine aggregates mainly due to their size, high sinking velocity and the micro-environment that they form (Jansen et al. 2002; Moriceau et al. 2007). Applying a Stokes' parameterization to the particles sinking through the mesopelagic and deeper could be more relevant than in the surface ocean. The acquisition of Stokes' like properties at depth through a loss of POM could also apply to aggregates that are ballasted only through their load of mineralizing phytoplankton cells. This is supported by several field studies that documented an increase of aggregate sinking speed with depth (Berelson 2002; Trull et al. 2008; Villa-Alfageme et al. 2016) and biogeochemical models include parameterizations of SV increase with depth to match the observed POM concentration profiles (Aumont et al. 2015; Gehlen et al. 2006; Kriest and Oschlies 2008). This is to some extent what our experiment reproduced. The drastic decrease of the fraction of POM in aggregates in all treatments testifies to the degradation that occurred in the tanks over the course of the 8 days of incubation (Le Moigne et al. 2013). It is also indicated by the decrease in TEP content observed in all tanks (except for the P treatment) between days 4 and 8. Moreover, we
recorded a continuous increase of aggregate SV during the experiment (Table 4), suggesting a
possible increase in excess density due to changing ratios of low (organic) to high (mineral)
density components. However, even towards the end of the experiment, aggregates from the P
treatment still did not present Stokes' like properties and other studies reported that an organic
matter loss at depth was accompanied by an increase of aggregate porosity (McCave and Gross
1991) and reported particularly low SV not correlated with size (Diercks and Asper 1997). In this
case, the empty space created by a loss of organic matter by remineralization would be replaced
by water — with a preservation of the initial architecture of the aggregate — instead of a
rearrangement that would possibly increase aggregate compactness thus promoting the
acquisition of Stokes' like properties. The increase in TEP content between days 4 and 8
observed only in the P treatment could also suggest a substantial bacterial growth (Decho 1990)
leading to the production of low-density matter. Unfortunately, there are no available in situ data
on porosity and size-SV relationship for individual particles in the deep ocean (> 1000 m) to
draw any firm conclusions.

We identified how the addition of calcite and micro-zooplankton can influence particle aspect,
size, shape and structure. Based on our results, calcite and to a lesser extent micro-zooplankton
grazing are responsible for a significant increase in particle roundness and compactness, which
improve Stokes' predictions. As expected, the inclusion of calcite in our particles significantly
increased their SV, which in turn significantly modified their shape, especially for the fastest-
sinking aggregates. In fact, circularities of particles in tanks 11 and 12 (those for which the best
fitting to Stokes' Law was obtained) were unexpectedly low if compared to the PZ treatment.

Small tails were observed on these particles (Fig. 2g and h) and we attribute the loss in circularity to these small changes in particle shape. This result further suggests that roundness is necessary but not sufficient in the acquisition of Stokes' properties but that the level of compactness is likely to be the most important factor to consider when trying to predict the SV of marine aggregates with a Stokes' model.

In addition to its effect on particle physico-chemical properties, calcite significantly influenced aggregation and disaggregation dynamics. In particular, the most stable size range (or ‘the equilibrium size’) at which the transfers between small to large (during coagulation) or large to small (during fragmentation), was higher in the treatments that contained calcite. In other terms, phyto-aggregates that did not include calcite, fragmented into smaller aggregates than did those that included calcite. This is an interesting result as it contradicts the common view that minerals tend to induce the formation of smaller aggregates essentially via fragmentation processes (Engel et al. 2009; Hamm 2002; Iversen and Ploug 2010; Passow and De La Rocha 2006). Moreover, in the PM treatment between days 4 and 8, both small and large particles converged to one single size class (~1-1.5 mm) via aggregation and fragmentation processes. It is likely that the fragmentation process for the mineral-loaded aggregates was different than that for the phytoplankton-only aggregates and therefore had different impact on the size spectra. As already discussed above, fragmentation in mineral-ballasted aggregates is triggered when the POC:mineral ratio drops below a certain threshold because of a diminution of particle stickiness.
Particles removed from their aggregate as fragmentation progresses would then most probably be
among the last that assembled— and thus not very small particles — since their attachment
would be the least consolidated. This process may explain why the fragmentation in this case led
to the formation of particles with intermediate sizes. Alternatively, the fragmentation or break-up
of phytoplankton aggregates could be due to a POC loss that increased further its already high
porosity and its fragility in any part of its structure. Small particles detaching from the larger one
during fragmentation could then be of any size and potentially very small. These hypotheses are
not easily tested in the frame of our experiment but Engel et al. (2009) who observed the
formation of smaller aggregates with a calcifying strain of *Emiliana huxleyi* also reported
contrasted fragmentation mechanisms between ballasted and non-ballasted aggregates. They
attributed these differences to the role of protection that minerals could play against microbial
remineralization — a mechanism still debated (Armstrong et al. 2002; Iversen and Ploug 2010;
Iversen and Robert 2015) — and showed that aggregates with minerals could be better protected
against fragmentation because of a more preserved organic cement. If true, this mechanism could
suggest that ballast minerals would promote the formation of medium-sized particles with high
density thereby serving as efficient carriers of carbon to the deep ocean.
Concluding remarks: implications for biogeochemical model parameterizations

The initial goal of this study was to ease SV parameterization in biogeochemical models by bringing evidence that Stokes' Law could reliably be applied to a certain category of marine aggregates (i.e. ballasted aggregates). Our results suggest that the SV of aggregated POM containing minerals could be parameterized in numerical models as a simple function of size and composition (i.e. relative fractions of organic and inorganic components, and porosity) using a Stokes' model. Although attempting such parameterization was beyond the scope of this paper we provide the following recommendations: (i) A Stokes' model parameterized with a fractal porosity (Eq. 7) should be used preferentially (model 3) to account for aggregate density variations with size. A fractal number $D_3$ of ~1.8 appears to be the most appropriate value since it adequately reproduces the slope of the size-SV relationship for the numerous datasets of ballasted aggregates considered here. It is important to note that a large range of fractal dimensions have been previously observed for marine aggregates (Kilps 1993; Maggi 2007) and this recommendation of a constant $D_3$ value is only intended here to facilitate sinking velocity parameterizations in biogeochemical models. At constant $D_3$ and for a given particle size range, porosity influencing aggregate density (and thus SV) is a function of the empirical constant $a$. Variations of $a$ between 0.1 and 0.001 reproduced the observed range of SV for aggregates carrying between ~7 and 98% of minerals (mineral:DW ratio). This observation suggests that $a$ could be empirically scaled to aggregate composition (especially to its mineral content), but additional field and experimental data are required to find an adequate formulation. If successful,
this result may importantly support the ‘ballast hypothesis’ (Armstrong et al. 2002; Armstrong et al. 2009; Klaas and Archer 2002) currently debated over the absence of a clear relationship between particle SV and mineral load (Aumont et al. 2017; Lee et al. 2009).

(ii) The applicability of our findings is likely to improve with increasing depth in the ocean due to the increase of the mineral:POM ratio through remineralization. This result implies that a parameterization of particle SV with a Stokes' model could be more robust in the mesopelagic layer and deeper ocean. Increase of particle SV with depth (Berelson 2002; Trull et al. 2008; Villa-Alfageme et al. 2016) could thus be potentially simulated through compositional changes in addition to depth-varying size classes (SV strictly size-dependent; Kriest and Oschlies 2008) or a prescribed linear SV increase with depth (Aumont et al. 2015; Gehlen et al. 2006; Kriest and Oschlies 2013).
Acknowledgements

We thank H. Stibor, Oceanopolis Brest, B. Becker, E. Grosstefan, M. Le Goff, P. Pondaven, A. C. Baudoux, R. Corvaisier, A. Masson, C. Liourzou (IUEM), C. Labry (IFREMER) for technical support, discussions, and/or advice. We would also like to thank F. Maggi and one anonymous reviewer for their constructive comments and helpful insights. This work was supported by the CalMarO FP7 Marie Curie Initial Training Network (www.calmaro.eu, grant agreement no. 215157), Region Bretagne, EURO-BASIN (No. 264933), an Integrating Project of the European Community 7th Framework Program, and by the Australian Research Council through a Laureate awarded to Philip W. Boyd (FL160100131).
<figure><img src="image_3.png" /></figure>

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Figure captions

**Fig. 1.** Theoretical expectations for porosity and drag ($C_D$) variations with the equivalent spherical diameter (ESD) for fractal aggregates. (a) Porosity calculated as a function of ESD for different values of coefficient $a$ and fractal dimension $D_3$ using the formula proposed by Logan and Wilkinson (1990) and following Laurenceau-Cornec et al. (2015b). (b) Drag coefficient calculated as a function of size following Laurenceau-Cornec et al. (2015b) using a bisection algorithm to solve numerically the White function (White 1974) and the Stokes' assumption as a starting point ($Re_0 = 24 / C_{D0}$); the initial drag coefficient $C_{D0}$ was calculated at two 'specific drag conditions' with arbitrarily fixed ESDs and corresponding sinking velocities using the Navier-Stokes' drag equation (1: $ESD_0 = 0.5$ mm, $SV_0 = 50$ m $d^{-1}$; 2: $ESD_0 = 0.5$ mm, $SV_0 = 300$ m $d^{-1}$); solid fractions of the aggregate were set at 60 % Particulate Organic Matter ($\rho$ = 1060 kg m$^{-3}$) and 40 % biogenic silica ($\rho$ = 2000 kg m$^{-3}$); fractal porosity is calculated as in panel a with different values of $a$ and $D_3$. Grey lines represent $C_D$ obtained with the same calculations using Stokes' law.

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**Fig. 2.** Photographs of aggregates formed in the rolling tanks on day 4 (**a-d**) and 8 (**e-h**) in the different treatments. P: Phytoplankton; PZ: Phytoplankton + Micro-Zooplankton (rotifers); PM: Phytoplankton + Mineral (calcite); PMZ: Phytoplankton + Mineral + Micro-Zooplankton. Addition of mineral and micro-zooplankton influenced aggregate size, shape and general aspect. Most of the differences marked on day 4 seemed to attenuate on day 8. Note the formation of small tails on day 8 in the PM and PMZ treatments for which the highest sinking velocities were recorded (white arrows).

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**Fig. 3.** Evolution of aggregate size spectra over the course of the experiment in the four treatments. (a) P: Phytoplankton; (b) PZ: Phytoplankton + Micro-Zooplankton (rotifers); (c) PM: Phytoplankton + Mineral (calcite); (d) PMZ: Phytoplankton + Mineral + Micro-Zooplankton. Small bar plots in the top right corner of each panel indicate particle number variations in each of the 14 size classes (see Table 1 for details) from day 2 to 4 (light grey) and from day 4 to 8 (dark grey). Size classes at which particle transfer operates via aggregation and disaggregation processes are indicated in bold. Aggregation processes illustrated by fewer particles in the small size classes and more in the large ones took place between days 2 and 4 in all treatments (no data on day 2 in the P treatment for which visible aggregates formed later). Disaggregation revealed by more particles in the small size classes and fewer in the large ones occurred between days 4 and 8. Note the occurrence of both disaggregation and aggregation in the two time periods in the PM treatment.
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**Fig. 4.** Box plots of aggregate equivalent spherical diameter (ESD) (a), circularity (b), and boundary fractal number $D_B$ (c) in the four treatments on day 2, 4 and 8. P: Phytoplankton; PZ: Phytoplankton + Micro-Zooplankton (rotifers); PM: Phytoplankton + Mineral (calcite); PMZ: Phytoplankton + Mineral + Micro-Zooplankton. Largest ESD variations over the experiment duration were observed in the P treatment while aggregates formed in presence of mineral and micro-zooplankton were more uniform in size and presented the highest circularities and lowest $D_B$. Only particles larger than 1 mm ESD were included in the analysis which required particle volume inferred from images.
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**Fig. 5.** Sinking velocities (SV) versus equivalent spherical diameter (ESD) for aggregates formed in each tank of the four treatments and comparison with theoretical expectations from different parameterizations of Stokes' law and the general Navier-Stokes' drag equation. P: Phytoplankton; PZ: Phytoplankton + Micro-Zooplankton (rotifers); PM: Phytoplankton + Mineral (calcite); PMZ: Phytoplankton + Mineral + Micro-Zooplankton. (a) Model 1, Stokes' law with constant porosities of 0 % (dashed lines) and 99 % (solid lines). (b) Model 2, general Navier-Stokes' drag law with constant drags of 1 (dashed lines) and 5 (solid lines), and a constant porosity of 99 %. (c) Model 3, Stokes' law with a porosity scaled on a fractal geometry with coefficient *a* = 0.03 and *D*<sub>3</sub> = 1.4 (dashed lines) and *D*<sub>3</sub> = 1.8 (solid lines). (d) Model 4, general Navier Stokes' drag law with a porosity scaled on a fractal geometry with coefficient *a* = 0.03 and *D*<sub>3</sub> = 1.4 (solid lines) and 1.8 (dashed lines). See the text for details on drag calculation.
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**Fig. 6.** Sinking velocity (SV) versus equivalent spherical diameter (ESD) from various studies for natural and experimentally-made aggregates and comparison with different model parameterizations (dark orange lines indicate the best fit to the data). (a) Aggregates without added minerals and Navier-Stokes' drag law (model 4, see text for details) with different fractal numbers $D_3$ (black lines: extreme values, grey lines: intermediate values), coefficient $a$ (dashed lines: $a = 0.1$; solid lines: $a = 0.01$) and three initial drag conditions indicated on the figure by numbers 1, 2, and 3 (1: ESD$_0$ = 0.1 mm, SV$_0$ = 300 m d$^{-1}$; 2: ESD$_0$ = 0.5 mm, SV$_0$ = 50 m d$^{-1}$; 3: ESD$_0$ = 1 mm, SV$_0$ = 5 m d$^{-1}$). Slopes and offset of the data best fit are reproduced by the model via variations of aggregate fractal dimension, coefficient $a$ and initial drag conditions. a: this study; b: Kajihara (1971); c: Shanks and Trent (1980); d: Alldredge and Gotschalk (1988); e: Alldredge and Gotschalk (1989); f-g: Azetsu-Scott and Johnson (1992), Bedford Basin and Diatom culture respectively; h: Diercks and Asper (1997); i: Ploug et al. (1999), one outlier removed (ESD = 1.14 mm; SV = 172 m d$^{-1}$); j-m: Engel et al. (2009), Diatoms Adriatic Sea, non-calcifying *E. huxleyi* culture, Diatoms Baltic Sea, and *N. closterium* culture respectively; n-p: Iversen and Ploug (2010), *S. costatum*, mix of *E. huxleyi* and *S. costatum*, *E. huxleyi* respectively; q: Jouandet et al. (2011); Laurenceau-Cornec et al. (2015b); s-w: Iversen and Robert (2015), Diatoms North Sea at 48, 91, 156, 187, and 380 h of incubation. (b) Aggregates with added minerals (except Engel et al., 2009) and the Stokes' law (model 3, see text and Fig. 5 caption) with different $D_3$ (grey dashed lines: $D_3$ = 1.4; black solid lines: $D_3$ = 1.8; grey solid lines: $D_3$ = 2.2) and coefficient $a$ (three groups of three grey dashed, black solid and grey solid lines with highest: $a$ = 0.1; intermediate: $a$ = 0.01; lowest: $a$ = 0.001). Slopes and offset of data best fit are well
reproduced by Stokes' law parameterized with D₃ values around 1.8 and coefficient a varying
between 0.1 and 0.001. a: this study; b: Carder et al. (1982); c: Gibbs (1985); d: Syvitski et al.
(1995); e: Nowald et al. (2009); f: Engel et al. (2009); g: Iversen et al. (2010); h-l: Iversen
and Robert (2015), Diatoms North Sea + Smectite at 48, 91, 156, 187, and 380 h of
incubation.

Author Manuscript
TABLES

<table><caption>**Table 1.** Limits and centre of the size classes used to bin the aggregates.</caption><thead><tr><th>Bins (cm)</th><th>1</th><th>2</th><th>3</th><th>4</th><th>5</th><th>6</th><th>7</th><th>8</th><th>9</th><th>10</th><th>11</th><th>12</th><th>13</th><th>14</th><th>15</th></tr></thead><tbody><tr><th>Lower limits (ESD)</th><td>0.0087</td><td>0.0125</td><td>0.0179</td><td>0.0255</td><td>0.0365</td><td>0.0521</td><td>0.0745</td><td>0.1065</td><td>0.1522</td><td>0.2176</td><td>0.311</td><td>0.4446</td><td>0.6354</td><td>0.9082</td><td>1.2981</td></tr><tr><th>Centre (ESD)</th><td>0.0106</td><td>0.0152</td><td>0.0217</td><td>0.031</td><td>0.0443</td><td>0.0633</td><td>0.0905</td><td>0.1294</td><td>0.1849</td><td>0.2643</td><td>0.3778</td><td>0.54</td><td>0.7718</td><td>1.1031</td><td>-</td></tr><tr><th>Upper limit (ESD)</th><td>0.0125</td><td>0.0179</td><td>0.0255</td><td>0.0365</td><td>0.0521</td><td>0.0745</td><td>0.1065</td><td>0.1522</td><td>0.2176</td><td>0.311</td><td>0.4446</td><td>0.6354</td><td>0.9082</td><td>1.2981</td><td>-</td></tr></tbody></table>
<table><caption>**Table 2.** Number, volume, organic and inorganic solid components, porosity (P) and excess density (Δρ) of all aggregates larger than 1 mm ESD.</caption><thead><tr><th>Treat.</th><th>Tank</th><th>Day</th><th>Agg #</th><th>Tot. agg. vol. (µL)</th><th>Tot. POM (mg)</th><th>Tot. PIC (mg)</th><th>Tot. bSiO<sub>2</sub> (mg)</th><th>Tot. TEP (mg XG eq.)</th><th>%POM (m:m<sub>sol</sub>)<sup>(1)</sup></th><th>%PIC (m:m<sub>sol</sub>)</th><th>%bSiO<sub>2</sub> (m:m<sub>sol</sub>)</th><th>P (% H<sub>2</sub>O) (v:v<sub>agg</sub>)<sup>(2)</sup></th><th>Δρ (kg m<sup>-3</sup>)</th></tr></thead><tbody><tr><td rowspan="3">P</td><td>1</td><td>2</td><td>0</td><td>0</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>5</td><td>4</td><td>81</td><td>1036.4±103.6</td><td>1.69±0.29</td><td>0.128</td><td>0.266±0.018</td><td>12.68±0.31</td><td>81.1±18.2</td><td>6.1±0.9</td><td>12.8±2</td><td>99.83±0.11</td><td>0.3</td></tr><tr><td>9</td><td>8</td><td>27</td><td>866.4±86.6</td><td>2.11±0.03</td><td>0.097±0.003</td><td>1.39±0.038</td><td>25.57±1.86</td><td>58.6±1.1</td><td>2.7±0.1</td><td>38.7±1.2</td><td>99.69±0.11</td><td>0.9±0.1</td></tr><tr><td rowspan="3">PZ</td><td>2</td><td>2</td><td>43</td><td>75.7±7.6</td><td>0.44</td><td>0.03</td><td>0.098</td><td>3.91</td><td>77.6</td><td>5.2</td><td>17.2</td><td>99.37±0.11</td><td>1.1±0.1</td></tr><tr><td>6</td><td>4</td><td>230</td><td>861.5±86.2</td><td>3.17</td><td>0.178±0.011</td><td>1.342±0.053</td><td>25±1.83</td><td>67.6±0.8</td><td>3.8±0.2</td><td>28.6±1.2</td><td>99.57±0.11</td><td>1±0.1</td></tr><tr><td>10</td><td>8</td><td>108</td><td>481.3±48.1</td><td>2.55±0.1</td><td>0.133±0.016</td><td>1.31±0.034</td><td>20.7±2.22</td><td>63.8±3</td><td>3.3±0.4</td><td>32.8±1.2</td><td>99.35±0.11</td><td>1.7±0.2</td></tr><tr><td rowspan="3">PM</td><td>3</td><td>2</td><td>226</td><td>368.4±36.8</td><td>1.38±0.08</td><td>1.376±0.047</td><td>0.259±0.097</td><td>13.5±0.8</td><td>45.8±3.3</td><td>45.6±2.5</td><td>8.6±3.2</td><td>99.47±0.11</td><td>2.8±0.4</td></tr><tr><td>7</td><td>4</td><td>210</td><td>1178.5±117.9</td><td>2.8±0.14</td><td>4.507</td><td>0.616±0.044</td><td>42.23±0.08</td><td>35.3±1.9</td><td>56.9±1.1</td><td>7.8±0.6</td><td>99.61±0.11</td><td>2.7±0.4</td></tr><tr><td>11</td><td>8</td><td>133</td><td>800.6±80.1</td><td>2.64±0.12</td><td>6.248±0.906</td><td>1.195±0.064</td><td>27.53±1.55</td><td>26.1±2.6</td><td>62±10.6</td><td>11.9±1.2</td><td>99.33±0.11</td><td>5.7±0.8</td></tr><tr><td rowspan="3">PMZ</td><td>4</td><td>2</td><td>211</td><td>465.9±46.6</td><td>1.95±0.17</td><td>1.697±0.036</td><td>0.388±0.015</td><td>9.77±0.71</td><td>48.3±4.8</td><td>42.1±2.1</td><td>9.6±0.6</td><td>99.43±0.11</td><td>2.8±0.4</td></tr><tr><td>8</td><td>4</td><td>196</td><td>1382.3±138.2</td><td>3.25±0.19</td><td>4.916±0.214</td><td>0.771±0.013</td><td>28.62</td><td>36.4±2.5</td><td>55±3</td><td>8.6±0.3</td><td>99.62±0.11</td><td>2.6±0.4</td></tr><tr><td>12</td><td>8</td><td>135</td><td>699.9±70</td><td>2.48±0.16</td><td>4.873±0.05</td><td>1.279±0.054</td><td>27.35±1.54</td><td>28.7±1.9</td><td>56.5±1.3</td><td>14.8±0.7</td><td>99.32±0.11</td><td>5.3±0.7</td></tr></tbody></table>

<footnote>(1) percentage of component in total solid content (m<sub>sol</sub>) taken as the sum of POM, PIC and bSiO<sub>2</sub> masses.</footnote>

<footnote>(2) P = volume of seawater in total aggregate volume (v<sub>agg</sub>).</footnote>
<table><caption>**Table 3.** Statistical results of ANOVAs, Kruskal-Wallis and Tukey tests for ESD, circularity and *D*<sub>B</sub> differences between treatments.</caption><thead><tr><th colspan="5">ANOVA</th><th rowspan="2">p-value</th><th colspan="2">Kruskal-Wallis</th><th rowspan="2">Tukey <sup>(2)</sup><br>p-value</th></tr><tr><th>Day</th><th>P</th><th>PZ</th><th>PM</th><th>PMZ</th><th>χ<sup>2</sup></th><th>p-value</th></tr></thead><tbody><tr><td colspan="9" style="text-align: left; font-weight: bold;">Equivalent Spherical Diameter ESD</td></tr><tr><td>2</td><td>-</td><td>1.34;<br>1.5±0.44<br>(43)</td><td>1.45;<br>1.49±0.3<br>(226)</td><td>1.57;<br>1.64±0.44<br>(211)</td><td>&lt;0.001</td><td>13.23</td><td>0.0013</td><td>4: 0.987<br>5: 0.071</td></tr><tr><td>4</td><td>1.91;<br>2.53±1.28<br>(81)</td><td>1.92;<br>1.93±0.51<br>(230)</td><td>2.17;<br>2.21±0.56<br>(210)</td><td>2.32;<br>2.35±0.67<br>(196)</td><td>&lt;0.001</td><td>47</td><td>&lt;0.001</td><td>2: 0.003<br>3: 0.217<br>6: 0.162</td></tr><tr><td>8<sup>(3)</sup></td><td>1.39;<br>1.63±0.69<br>(26)</td><td>1.98;<br>1.99±0.56<br>(108)</td><td>1.98;<br>2.17±0.72<br>(133)</td><td>1.9;<br>2.03±0.74<br>(135)</td><td>0.003</td><td>16.23</td><td>0.001</td><td>1: 0.084<br>2: 0.002<br>3: 0.036<br>4: 0.169<br>5: 0.966<br>6: 0.328</td></tr><tr><td colspan="9" style="text-align: left; font-weight: bold;">Circularity</td></tr><tr><td>2</td><td>-</td><td>0.543;<br>0.548±0.13<br>(26)</td><td>0.779;<br>0.75±0.12<br>(105)</td><td>0.666;<br>0.66±0.14<br>(61)</td><td>&lt;0.001</td><td>45.97</td><td>&lt;0.001</td><td>all &lt;0.001</td></tr><tr><td>4</td><td>0.461;<br>0.465±0.12<br>(45)</td><td>0.578;<br>0.567±0.14<br>(118)</td><td>0.668;<br>0.656±0.113<br>(73)</td><td>0.652;<br>0.647±0.11<br>(44)</td><td>&lt;0.001</td><td>61.23</td><td>&lt;0.001</td><td>5: 0.002<br>6: 0.98</td></tr><tr><td>8</td><td>0.504;<br>0.478±0.11<br>(20)</td><td>0.75;<br>0.714±0.13<br>(69)</td><td>0.6;<br>0.594±0.15<br>(62)</td><td>0.659;<br>0.658±0.13<br>(64)</td><td>&lt;0.001</td><td>47.82</td><td>&lt;0.001</td><td>2: 0.004<br>5: 0.08<br>6: 0.037</td></tr><tr><td colspan="9" style="text-align: left; font-weight: bold;">Boundary fractal number D<sub>B</sub></td></tr><tr><td>2</td><td>-</td><td>1.113;<br>1.117±0.03<br>(26)</td><td>1.068;<br>1.073±0.03<br>(103)</td><td>1.08;<br>1.084±0.03<br>(060)</td><td>&lt;0.001</td><td>36.7</td><td>&lt;0.001</td><td>6: 0.046</td></tr><tr><td>4</td><td>1.13;<br>1.138±0.04<br>(45)</td><td>1.107;<br>1.114±0.04<br>(118)</td><td>1.082;<br>1.085±0.03<br>(73)</td><td>1.09;<br>1.092±0.03<br>(42)</td><td>&lt;0.001</td><td>58.23</td><td>&lt;0.001</td><td>1: 0.026<br>5: 0.003<br>6: 0.79</td></tr><tr><td>8</td><td>1.128;<br>1.128±0.03<br>(20)</td><td>1.069;<br>1.08±0.03<br>(68)</td><td>1.101;<br>1.103±0.03<br>(61)</td><td>1.09;<br>1.09±0.03<br>(63)</td><td>&lt;0.001</td><td>41.74</td><td>&lt;0.001</td><td>2: 0.0164<br>5: 0.147<br>6: 0.158</td></tr></tbody></table>

<footnote>(1) X̄: median; X̄: mean; SD: standard deviation to the mean; n: number of observations.</footnote>

<footnote>(2) Only *p*-values > 0.001 are indicated (*p*-values of all other pairwise comparisons < 0.001); Pairwise comparisons between treatments: 1=PZ-P, 2=PM-P, 3=PMZ-P, 4=PM-PZ, 5=PMZ-PZ, 6=PMZ-PM.</footnote>

<footnote>(3) One outlier with an ESD larger than 12 mm removed from the analysis.</footnote>
<table><caption>**Table 4.** Average values, standard deviations (SD), minima and maxima of aggregate equivalent spherical diameters (ESD), sinking velocities (SV), Reynolds numbers (Re), drag coefficient ($C_D$) are indicated for each tank. Coefficients and statistics of the power regression fits (SV = a . ESD$^b$) to the observations are also reported. The number of aggregates sub-sampled from all particles for sinking velocity measurements in each tank is indicated (n).</caption><thead><tr><th>Treat.</th><th>Tank (n)</th><th><span style="text-decoration: overline;">ESD</span>±SD (mm)<br>[min., max.]</th><th><span style="text-decoration: overline;">SV</span>±SD (m d<sup>-1</sup>)<br>[min., max.]</th><th><span style="text-decoration: overline;">Re</span>±SD<br>[min., max.]</th><th><span style="text-decoration: overline;">C<sub>D</sub>s</span> ± SD<sup>(1)</sup><br>[min., max.]</th><th><span style="text-decoration: overline;">C<sub>D</sub>w</span> ± SD<sup>(2)</sup><br>[min., max.]</th><th>a; SE; p-value<br>[95% CI]<sup>(3)</sup></th><th>b; SE; p-value<br>[95% CI]</th></tr></thead><tbody><tr><td rowspan="2">P</td><td>5 (11)</td><td>3.18±0.52<br>[1.99, 4]</td><td>361±76<br>[225, 493]</td><td>11.4±3.6<br>[4.4, 17.3]</td><td>2.4±1.1<br>[1.4, 5.5]</td><td>4.2±1.3<br>[3, 7.8]</td><td>159; 68.7; &lt;0.05<br>[55.7, 378.8]</td><td>0.711; 0.36; 0.082<br>[-0.034, 1.582]</td></tr><tr><td>9 (33)</td><td>4.61±1.64<br>[1.92, 7.51]</td><td>511±125<br>[218, 928]</td><td>22.6±8.6<br>[5.7, 43.8]</td><td>1.3±0.7<br>[0.5, 4.2]</td><td>2.8±0.9<br>[1.7, 6.4]</td><td>667.1; 103.1; &lt;0.001<br>[474.4, 914.4]</td><td>-0.184; 0.1; 0.093<br>[-0.408, 0.043]</td></tr><tr><td rowspan="3">PZ</td><td>2 (6)</td><td>1.43±0.27<br>[1.16, 1.78]</td><td>276±37<br>[234, 317]</td><td>3.9±1.2<br>[2.6, 5.5]</td><td>6.6±1.9<br>[4.4, 9.1]</td><td>9.1±2.1<br>[6.6, 11.7]</td><td>228.9; 22.3; &lt;0.001<br>[171.3, 296.3]</td><td>0.539; 0.24; 0.087<br>[-0.137, 1.205]</td></tr><tr><td>6 (6)</td><td>3.61±0.89<br>[2.44, 4.6]</td><td>404±114<br>[226, 543]</td><td>14±4.9<br>[9.4, 22.4]</td><td>1.9±0.6<br>[1.1, 2.6]</td><td>3.6±0.7<br>[2.5, 4.4]</td><td>706.3; 433.9; 0.179<br>[-498.4, 1911]</td><td>-0.45; 0.5; 0.419<br>[-1.836, 0.9561]</td></tr><tr><td>10 (16)</td><td>4.34±0.86<br>[3.1, 5.92]</td><td>493±132<br>[311, 862]</td><td>21.3±8.5<br>[10.9, 43.4]</td><td>1.3±0.5<br>[0.6, 2.2]</td><td>2.8±0.7<br>[1.7, 4]</td><td>200.7; 97.9; 0.059<br>[69.1, 543.5]</td><td>0.615; 0.32; 0.078<br>[-0.063, 1.31]</td></tr><tr><td rowspan="3">PM</td><td>3 (5)</td><td>1.07±0.34<br>[0.57, 1.49]</td><td>397±60<br>[327, 461]</td><td>4.3±1.8<br>[2, 6.7]</td><td>6.8±3.4<br>[3.6, 12.1]</td><td>9.2±3.7<br>[5.7, 15]</td><td>393.6; 22.7; &lt;0.001<br>[320.4, 465.6]</td><td>0.3; 0.19; 0.205<br>[-0.265, 0.99]</td></tr><tr><td>7 (17)</td><td>1.5±0.37<br>[1.01, 2.38]</td><td>833±184<br>[555, 1305]</td><td>12.7±6.3<br>[6.7, 30.3]</td><td>2.2±0.8<br>[0.8, 3.6]</td><td>4±1<br>[2.1, 5.7]</td><td>617.5; 43.7; &lt;0.001<br>[526.9, 716.8]</td><td>0.742; 0.13; &lt;0.001<br>[0.442, 1.031]</td></tr><tr><td>11 (49)</td><td>1.98±0.37<br>[1.24, 2.8]</td><td>1181±256<br>[698, 1704]</td><td>23.4±8.7<br>[10.7, 44.4]</td><td>1.2±0.4<br>[0.5, 2.3]</td><td>2.6±0.6<br>[1.7, 4.1]</td><td>722.7; 72.2; &lt;0.001<br>[585.7, 884]</td><td>0.723; 0.13; &lt;0.001<br>[0.445, 1]</td></tr><tr><td>PMZ</td><td>4 (5)</td><td>1.68±0.49<br>[1.09, 2.32]</td><td>724±149<br>[543, 941]</td><td>12.3±5.7<br>[5.8, 21.3]</td><td>2.3±1.1<br>[1.1, 4.1]</td><td>4.1±1.4<br>[2.6, 6.3]</td><td>563.7; 100; &lt;0.05<br>[280.2, 916.6]</td><td>0.497; 0.29; 0.186<br>[-0.411, 1.527]</td></tr></tbody></table>
<table><tr><td>8 (22)</td><td>1.82±0.43<br>[0.72, 2.63]</td><td>896±254<br>[352, 1385]</td><td>16.7±7.5<br>[3.1, 35]</td><td>2±1.9<br>[0.7, 7.8]</td><td>3.7±2.2<br>[2, 10.4]</td><td>519.5; 72.1; &lt;0.001<br>[377, 678.4]</td><td>0.911; 0.2; &lt;0.001<br>[0.517, 1.362]</td></tr><tr><td>12 (33)</td><td>2.08±0.43<br>[1.41, 3]</td><td>1201±233<br>[759, 1697]</td><td>25±9.4<br>[11.6, 49.6]</td><td>1.1±0.4<br>[0.5, 2.1]</td><td>2.5±0.6<br>[1.6, 3.8]</td><td>763.6; 76.7; &lt;0.001<br>[616.7, 935.5]</td><td>0.622; 0.13; &lt;0.001<br>[0.361, 0.887]</td></tr></table>

<footnote>(1) $\overline{C}_D$ s: average drag coefficient calculated from the Reynolds number using the Stokes' assumption $C_D = 24/Re$.</footnote>

<footnote>(2) $\overline{C}_D$ w: average drag coefficient calculated from the Reynolds number using White's empirical relationship (White 1974) $C_D = 24/Re + 6/(1+Re^{0.5}) + 0.4$.</footnote>

<footnote>(3) Standard Error, *p*-value and 95 % confidence intervals.</footnote>
<table><caption>**Table 5.** Criteria for inter-comparisons of model performance to predict log-transformed observations. Bold values indicate the model pairwise inter-comparison: model 1 versus 3 for the treatment with calcite and model 2 versus 4 for the treatments without calcite (see text for details on model parameterization). Italicized values indicate the occurrence of a difference in log(Likelihood) larger than 2 points and taken as a criterion for a better fit.</caption><thead><tr><th rowspan="2">Treat.</th><th rowspan="2">Tank</th><th rowspan="2">n</th><th colspan="3">Model 1</th><th colspan="3">Model 2</th><th colspan="3">Model 3</th><th colspan="3">Model 4</th></tr><tr><th>RSS</th><th>RMSE</th><th>log (L)</th><th>RSS</th><th>RMSE</th><th>log (L)</th><th>RSS</th><th>RMSE</th><th>log (L)</th><th>RSS</th><th>RMSE</th><th>log (L)</th></tr></thead><tbody><tr><td rowspan="2">P</td><td>5</td><td>11</td><td>0.61</td><td>0.23</td><td>6.93</td><td><strong>0.12</strong></td><td><strong>0.1</strong></td><td><strong>10.86</strong></td><td>0.65</td><td>0.24</td><td>6.76</td><td><strong>0.1</strong></td><td><strong>0.09</strong></td><td><strong>11.3</strong></td></tr><tr><td>9</td><td>33</td><td>17.59</td><td>0.73</td><td>4.51</td><td><strong>0.72</strong></td><td><strong>0.15</strong></td><td><strong>27.45</strong></td><td>13.97</td><td>0.65</td><td>6.16</td><td><strong>0.39</strong></td><td><strong>0.11</strong></td><td><strong>31.74</strong></td></tr><tr><td rowspan="3">PZ</td><td>2</td><td>6</td><td>0.71</td><td>0.35</td><td>2.77</td><td><strong>0.08</strong></td><td><strong>0.12</strong></td><td><strong>5.61</strong></td><td>0.16</td><td>0.16</td><td>4.71</td><td><strong>0.02</strong></td><td><strong>0.05</strong></td><td><strong>7.79</strong></td></tr><tr><td>6</td><td>6</td><td>1.71</td><td>0.53</td><td>1.64</td><td><strong>0.14</strong></td><td><strong>0.15</strong></td><td><strong>4.92</strong></td><td>1.5</td><td>0.5</td><td>1.8</td><td><strong>0.09</strong></td><td><strong>0.12</strong></td><td><strong>5.46</strong></td></tr><tr><td>10</td><td>16</td><td>5.72</td><td>0.6</td><td>3.58</td><td><strong>0.14</strong></td><td><strong>0.09</strong></td><td><strong>16.54</strong></td><td>5.09</td><td>0.56</td><td>3.98</td><td><strong>0.2</strong></td><td><strong>0.11</strong></td><td><strong>15.24</strong></td></tr><tr><td rowspan="3">PM</td><td>3</td><td>5</td><td><strong>0.68</strong></td><td><strong>0.37</strong></td><td><strong>2.17</strong></td><td>0.06</td><td>0.11</td><td>4.88</td><td><strong>0.04</strong></td><td><strong>0.09</strong></td><td><strong>5.14</strong></td><td>0.02</td><td>0.06</td><td>6.15</td></tr><tr><td>7</td><td>17</td><td><strong>0.71</strong></td><td><strong>0.2</strong></td><td><strong>11.72</strong></td><td>1.29</td><td>0.28</td><td>9.53</td><td><strong>0.06</strong></td><td><strong>0.06</strong></td><td><strong>20.62</strong></td><td>0.13</td><td>0.09</td><td>17.9</td></tr><tr><td>11</td><td>49</td><td><strong>0.91</strong></td><td><strong>0.14</strong></td><td><strong>42.44</strong></td><td>5.15</td><td>0.32</td><td>23.97</td><td><strong>1.31</strong></td><td><strong>0.16</strong></td><td><strong>38.52</strong></td><td>0.5</td><td>0.1</td><td>48.86</td></tr><tr><td rowspan="3">PMZ</td><td>4</td><td>5</td><td><strong>0.28</strong></td><td><strong>0.24</strong></td><td><strong>3.12</strong></td><td>0.36</td><td>0.27</td><td>2.87</td><td><strong>0.02</strong></td><td><strong>0.07</strong></td><td><strong>5.87</strong></td><td>0.03</td><td>0.08</td><td>5.45</td></tr><tr><td>8</td><td>22</td><td><strong>0.58</strong></td><td><strong>0.16</strong></td><td><strong>17.4</strong></td><td>1.64</td><td>0.27</td><td>12.39</td><td><strong>0.4</strong></td><td><strong>0.13</strong></td><td><strong>19.15</strong></td><td>0.43</td><td>0.14</td><td>18.78</td></tr><tr><td>12</td><td>33</td><td><strong>0.67</strong></td><td><strong>0.14</strong></td><td><strong>27.88</strong></td><td>3.74</td><td>0.34</td><td>15.61</td><td><strong>0.74</strong></td><td><strong>0.15</strong></td><td><strong>27.26</strong></td><td>0.28</td><td>0.09</td><td>34.31</td></tr></tbody></table>

RSS: Residual sum of squares; RMSE: Root Mean Square Error; L: Likelihood.
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<figure><img src="image_11.png" /><figcaption>FIGURE 1</figcaption></figure>

**Fig. 1.** Theoretical expectations for porosity and drag ($C_D$) variations with the equivalent spherical diameter (ESD) for fractal aggregates. (a) Porosity calculated as a function of ESD for different values of coefficient $a$ and fractal dimension $D_3$ using the formula proposed by Logan and Wilkinson (1990) and following Laurenceau-Cornec et al. (2015b). (b) Drag coefficient calculated as a function of size following Laurenceau-Cornec et al. (2015b) using a bisection algorithm to solve numerically the White function (White 1974) and the Stokes' assumption as a starting point ($Re_0 = 24 / C_{D0}$); the initial drag coefficient $C_{D0}$ was calculated at two 'specific drag conditions' with arbitrarily fixed ESDs and corresponding sinking velocities using the Navier-Stokes' drag equation (1: $ESD_0 = 0.5$ mm, $SV_0 = 50$ m $d^{-1}$; 2: $ESD_0 = 0.5$ mm, $SV_0 = 300$ m $d^{-1}$); solid fractions of the aggregate were set at 60 % Particulate Organic Matter ($\rho$ = 1060 kg m$^{-3}$) and 40 % biogenic silica ($\rho$ = 2000 kg m$^{-3}$); fractal porosity is calculated as in panel a with different values of $a$ and $D_3$. Grey lines represent $C_D$ obtained with the same calculations using Stokes' law.
<figure><img src="image_12.png" /><figcaption>FIGURE 2</figcaption></figure>

**Fig. 2.** Photographs of aggregates formed in the rolling tanks on day 4 (a-d) and 8 (e-h) in the different treatments. P: Phytoplankton; PZ: Phytoplankton + Micro-Zooplankton (rotifers); PM: Phytoplankton + Mineral (calcite); PMZ: Phytoplankton + Mineral + Micro-Zooplankton. Addition of mineral and micro-zooplankton influenced aggregate size, shape and general aspect. Most of the differences marked on day 4 seemed to attenuate on day 8. Note the formation of small tails on day 8 in the PM and PMZ treatments for which the highest sinking velocities were recorded (white arrows).
<figure><img src="image_13.png" /><figcaption>FIGURE 3</figcaption></figure>

**Fig. 3.** Evolution of aggregate size spectra over the course of the experiment in the four treatments. (a) P: Phytoplankton; (b) PZ: Phytoplankton + Micro-Zooplankton (rotifers); (c) PM: Phytoplankton + Mineral (calcite); (d) PMZ: Phytoplankton + Mineral + Micro-Zooplankton. Small bar plots in the top right corner of each panel indicate particle number variations in each of the 14 size classes (see Table 1 for details) from day 2 to 4 (light grey) and from day 4 to 8 (dark grey). Size classes at which particle transfer operates via aggregation and disaggregation processes are indicated in bold. Aggregation processes
illustrated by fewer particles in the small size classes and more in the large ones took place
between days 2 and 4 in all treatments (no data on day 2 in the P treatment for which visible
aggregates formed later). Disaggregation revealed by more particles in the small size classes
and fewer in the large ones occurred between days 4 and 8. Note the occurrence of both
disaggregation and aggregation in the two time periods in the PM treatment.

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<figure><img src="image_14.png" /><figcaption>FIGURE 4</figcaption></figure>

**Fig. 4.** Box plots of aggregate equivalent spherical diameter (ESD) (a), circularity (b), and boundary fractal number $D_B$ (c) in the four treatments on day 2, 4 and 8. P: Phytoplankton; PZ: Phytoplankton + Micro-Zooplankton (rotifers); PM: Phytoplankton + Mineral (calcite); PMZ: Phytoplankton + Mineral + Micro-Zooplankton. Largest ESD variations over the experiment duration were observed in the P treatment while aggregates formed in presence of mineral and micro-zooplankton were more uniform in size and presented the highest circularities and lowest $D_B$. Only particles larger than 1 mm ESD were included in the analysis which required particle volume inferred from images.
<figure><img src="image_15.png" /><figcaption>FIGURE 5</figcaption></figure>

**Fig. 5.** Sinking velocities (SV) versus equivalent spherical diameter (ESD) for aggregates formed in each tank of the four treatments and comparison with theoretical expectations from different parameterizations of Stokes' law and the general Navier-Stokes' drag equation. P: Phytoplankton; PZ: Phytoplankton + Micro-Zooplankton (rotifers); PM: Phytoplankton + Mineral (calcite); PMZ: Phytoplankton + Mineral + Micro-Zooplankton. (a) Model 1, Stokes' law with constant porosities of 0 % (dashed lines) and 99 % (solid lines). (b) Model 2,
general Navier-Stokes' drag law with constant drags of 1 (dashed lines) and 5 (solid lines), and a constant porosity of 99 %. (c) Model 3, Stokes' law with a porosity scaled on a fractal geometry with coefficient *a* = 0.03 and *D*<sub>3</sub> = 1.4 (dashed lines) and *D*<sub>3</sub> = 1.8 (solid lines). (d) Model 4, general Navier Stokes' drag law with a porosity scaled on a fractal geometry with coefficient *a* = 0.03 and *D*<sub>3</sub> = 1.4 (solid lines) and 1.8 (dashed lines). See the text for details on drag calculation.

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<figure><img src="image_16.png" /><figcaption>FIGURE 6</figcaption></figure>

**Fig. 6.** Sinking velocity (SV) versus equivalent spherical diameter (ESD) from various studies for natural and experimentally-made aggregates and comparison with different model parameterizations (dark orange lines indicate the best fit to the data). (a) Aggregates without added minerals and Navier-Stokes' drag law (model 4, see text for details) with different fractal numbers $D_3$ (black lines: extreme values, grey lines: intermediate values), coefficient $a$ (dashed lines: $a = 0.1$; solid lines: $a = 0.01$) and three initial drag conditions indicated on the figure by numbers 1, 2, and 3 (1: ESD$_0$ = 0.1 mm, SV$_0$ = 300 m d$^{-1}$; 2: ESD$_0$ = 0.5 mm, SV$_0$ = 50 m d$^{-1}$; 3: ESD$_0$ = 1 mm, SV$_0$ = 5 m d$^{-1}$). Slopes and offset of the data best fit are reproduced by the model via variations of aggregate fractal dimension, coefficient $a$ and initial drag conditions. a: this study; b: Kajihara (1971); c: Shanks and Trent (1980); d: Alldredge and Gotschalk (1988); e: Alldredge and Gotschalk (1989); f-g: Azetsu-Scott and Johnson (1992), Bedford Basin and Diatom culture respectively; h: Diercks and Asper (1997); i: Ploug et al. (1999), one outlier removed (ESD = 1.14 mm; SV = 172 m d$^{-1}$); j-m:
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Engel et al. (2009), Diatoms Adriatic Sea, non-calcifying *E. huxleyi* culture, Diatoms Baltic Sea, and *N. closterium* culture respectively; n-p: Iversen and Ploug (2010), *S. costatum*, mix of *E. huxleyi* and *S. costatum*, *E. huxleyi* respectively; q: Jouandet et al. (2011); Laurenceau-Cornec et al. (2015b); s-w: Iversen and Robert (2015), Diatoms North Sea at 48, 91, 156, 187, and 380 h of incubation. (b) Aggregates with added minerals (except Engel et al., 2009) and the Stokes' law (model 3, see text and Fig. 5 caption) with different *D*₃ (grey dashed lines: *D*₃ = 1.4; black solid lines: *D*₃ = 1.8; grey solid lines: *D*₃ = 2.2) and coefficient *a* (three groups of three grey dashed, black solid and grey solid lines with highest: *a* = 0.1; intermediate: *a* = 0.01; lowest: *a* = 0.001). Slopes and offset of data best fit are well reproduced by Stokes' law parameterized with *D*₃ values around 1.8 and coefficient *a* varying between 0.1 and 0.001. a: this study; b: Carder et al. (1982); c: Gibbs (1985); d: Syvitski et al. (1995); e: Nowald et al. (2009); f: Engel et al. (2009); g: Iversen et al. (2010); h-l: Iversen and Robert (2015), Diatoms North Sea + Smectite at 48, 91, 156, 187, and 380 h of incubation.
New guidelines for the application of Stokes' models
to the sinking velocity of marine aggregates

Emmanuel C. Laurenceau-Cornec¹a*, Frédéric A. C. Le Moigne²,⁵b, Morgane Gallinari¹, Brivaela

Moriceau¹, Jordan Toullec¹, Morten Iversen³,⁴, Anja Engel⁵, Christina L. De La Rocha¹.

¹ Univ Brest, CNRS, IRD, Ifremer, LEMAR, F-29280 Plouzane, France.

² National Oceanography Centre, Southampton, European Way, SO14 3ZH, UK.

³ Alfred Wegener Institute for Polar and Marine Research, Bremerhaven, Germany.

⁴ MARUM and University of Bremen, Bremen, Germany.

⁵ GEOMAR, Helmholtz Centre for Ocean Research Kiel, Düsternbrooker Weg 20, D-24105 Kiel, Germany.

ª Now at Institute for Marine and Antarctic Studies, University of Tasmania 7001 AUSTRALIA.

ᵇ Now at Mediterranean Institute of Oceanography (MIO), UMR CNRS 7294 - IRD 235, UM AMU 110, 163 Avenue de Luminy, 13288 Marseille Cedex 09.

Emmanuel Laurenceau-Cornec (*corresponding author): emmanuel.laurenceau@utas.edu.au

Frédéric Le Moigne: frederic.lemoigne@mio.osupytheas.fr

Morgane Gallinari: Morgane.Gallinari@univ-brest.fr

Brivaela Moriceau: brivaela.moriceau@univ-brest.fr

Jordan Toullec: jordan.toullec@univ-brest.fr

Morten Iversen: miversen@marum.de

Anja Engel: aengel@geomar.de

Christina De La Rocha: xtinadlr@hushmail.com

Keywords : marine snow; Stokes’ Law; mineral ballast; sinking velocity; aggregation;

zooplankton.

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31 Abstract

Numerical simulations of ocean biogeochemical cycles need to adequately represent particle sinking velocities (SV). For decades, Stokes' Law estimating particle SV from density and size has been widely used. But while Stokes' Law holds for small, smooth and rigid spheres settling at low Reynolds number, it fails when applied to marine aggregates complex in shape, structure and composition.

Minerals and zooplankton can alter phytoplankton aggregates in ways that change their SV, potentially improving the applicability of Stokes' models. Using rolling cylinders, we experimentally produced diatom aggregates in the presence and absence of minerals and/or micro-zooplankton. Minerals and to a lesser extent micro-zooplankton decreased aggregate size and roughness and increased their sphericity and compactness. Stokes' Law parameterized with a fractal porosity modelled adequately size-SV relationships for mineral-loaded aggregates.

Phytoplankton-only aggregates and those exposed to micro-zooplankton followed the general Navier-Stokes drag equation suggesting an indiscernible effect of micro-zooplankton and a drag coefficient too complex to be calculated with a Stokes' assumption. We compared our results with a larger dataset of ballasted and non-ballasted marine aggregates. This confirmed that the size-SV relationships for ballasted aggregates can be simulated by Stokes' models with an adequate fractal porosity parameterization. Given the importance of mineral ballasting in the ocean, our findings could ease biogeochemical model parameterization for a significant pool of particles in the ocean and especially in the mesopelagic zone where the particulate organic matter:mineral ratio decreases. Our results also reinforce the importance of accounting for porosity as a decisive predictor of marine aggregate SV.
## Introduction

The settling of marine particulate matter despite being complex in shape, structure and density has traditionally been modelled as simple spherical particles in order to ease the numerical computation of the many biogeochemical processes in which sinking particles are involved. Since the 70's (Lerman et al. 1974), Stokes' Law (Stokes 1851) has been used to represent particle sinking velocities (SV) in ocean models. Stokes' Law is shown in equation 1,

$$U_s = \frac{g(\rho_s - \rho_f)D^2}{18\mu} \qquad (1)$$

where $U_s$ is the SV of a sphere in m s⁻¹, g is the acceleration due to gravity (9.81 m s⁻²), $\rho_s$ and $\rho_f$ are the sphere and fluid densities in kg m⁻³, D is the sphere diameter in m and $\mu$ is the fluid dynamic viscosity in kg m⁻¹ s⁻¹. Stokes' Law is a solution of the low dimensionless Reynolds number (Reynolds 1883) limit of the Navier-Stokes equation. Initially, it was formulated for small perfect spheres settling in a quiescent fluid at low velocity, assuming that inertial forces are negligible compared to viscous forces. It was empirically found to be valid at low Reynolds number $Re < 0.5$,

$$Re = \frac{U_s D}{\nu} \qquad (2)$$

where $\nu$ is the seawater kinematic viscosity in m² s⁻¹. Stokes' Law conveniently allows estimation of the SV of a sphere based on density and diameter only, two particle properties readily measured using in situ imaging and chemical analyses. The SV of natural marine particles such as phytoplankton cells well characterized in term of geometry and composition can potentially be modelled by Stokes' Law (Waite et al. 1992) or empirically modified versions (Miklasz and Denny 2010) given small variations in cell shape and physiological state affecting its density (Smayda 1970). However, large marine aggregates (e.g. marine snow) are complex in

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shape and structure compared to the perfect spheres in Stokes' equation (Chase 1979). For natural large marine aggregates, the following general Navier-Stokes' drag equation (Eq. 3; Stokes, 1851) can be used to estimate SV,

where $U_{agg}$ is the aggregate SV in m s⁻¹, $Δρ$ is the excess density in kg m⁻³ (= the difference between the aggregate and fluid densities; equivalent to $ρ_s - ρ_f$ in Eq. 1), and $C_D$ is the dimensionless drag coefficient. The drag coefficient depends upon the shape and surface structure (i.e. roughness) of the particle and is well defined for those with known shapes such as fecal pellets' (tabular, cylindrical, conical, etc.). As a consequence, the SV of fecal pellets has been successfully resolved (Komar et al. 1981; Ploug et al. 2008; Yoon et al. 2001), and does not present major difficulties for model parameterization (e.g. Jackson 2001). However, the complex shape, structure and roughness of phyto-aggregates complicate estimations of the drag coefficient and thus the SV (Alldredge and Gotschalk 1988; Johnson et al. 1996; Loth 2008). Particle SV parameterizations in numerical ocean models tend to avoid a strict size-dependency. Instead, they use theoretically and empirically derived equations (Maggi 2013). These equations account for a large number of SV drivers such as particle shape (and the non-negligible drag associated), porosity, structure, fluid temperature and salinity (Jokulsdottir and Archer 2016; Kriest and Oschlies 2008). Complexity in simulation is necessary to mimic natural processes, but it also presents the huge inconvenience of assuming some of the many parameters that are poorly constrained by observation or experiments. Model results are then limited to a very specific experimental or biogeochemical context.

$$U_{agg} = \left[ \frac{4 g(Δρ)D}{3 C_D ρ_f} \right]^{1/2} \qquad (3)$$
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Density $\rho_s$ and diameter $D$ are the two most important particle characteristics accounted for in Stokes' Law calculations (Eq. 1). The latter is easily accessible by different imaging methods that can be used in situ or on experimental studies in the lab or at sea (e.g. Holographic Particle Velocimeter, Flow Cytometer And Microscope, Underwater Vision Profiler, Laser In Situ Scattering and Transmissometry, Laser Optical Particle Counter, Coulter Counter, light photography, etc.). In theory, aggregate density can be calculated from measurements of the relative mass fractions of the different solid components combined with measurements of particle dry weights and volumes to access the porosity (Eq. 1 & 3). In practice, individual and simultaneous measurements and calculations of both diameter and density on particles are rarely performed because of the difficulties they represent. Instead, these measurements are often made on a mixture of aggregates and rarely on individual particles due to the detection limits for very small amounts of solid components contained in a single particle. The fraction of aggregate volume occupied by seawater (i.e. the porosity) that also enters the calculations of aggregate density is inferred from measurements of aggregate volume or dry weight (Engel and Schartau 1999; Iversen and Ploug 2010; Laurenceau-Cornec et al. 2015b; Passow and De La Rocha 2006), or from a calculation based on aggregate cell concentration and sizes (assuming limited variations of cell sizes) and aggregate volume (Moriceau et al. 2007). For similar particles (i.e. formed in the same conditions and from the same pool of POM), the relative fractions of the solid components within each particle are not subjected to large variations from one particle to another. The chemical composition of a mixture of aggregates is thus likely to reflect the overall composition of each individual aggregate. However, particle porosity varies with size as marine aggregates are polydisperse systems (i.e. composed of particles of non-uniform sizes) and have a fractal nature (Logan and Wilkinson 1990; Meakin 1988). In essence, this means that the rate at
116 which the aggregate volume is filled with new particles during aggregation is lower than the rate
117 at which the size increases. The direct consequences are that aggregate porosity increases with
118 increasing particle size and thus large marine aggregates have a lower density than small
119 aggregates of similar composition. Parameterization of particle density in Stokes' Law should
120 thus account for these variations of particle density with size.

121 If Stokes' Law is to be valid for a certain type of marine particles, it is likely that these particles
122 are roughly spherical (or 'round'), limited in size and compact (low porosity). Minerals are well
123 known for their so-called 'ballast effect' which describes an increase of size-specific aggregate
124 excess density and SV upon mineral incorporation (Armstrong et al. 2002; Klaas and Archer
125 2002). Additionally, aggregation in the presence of minerals promote the formation of more
126 numerous, compact, spherical and smaller particles (Engel et al. 2009; Hamm 2002; Iversen and
127 Ploug 2010; Jagt et al. 2018; Nowald et al. 2015; Passow and De La Rocha 2006; Passow et al.
128 2014; Schmidt et al. 2014). By promoting fragmentation processes, minerals limit the formation
129 of large particles. This could be explained by a reduction of particle stickiness (i.e. the
130 probability that two particles assemble upon collision) in the presence of minerals. In aggregates
131 including organic and mineral matter, the POM acts as a glue which can lose its efficiency and
132 lead to disaggregation when the POC:DW mass ratio falls below a value around 3-7% 'called the
133 POC carrying capacity' which depends on the mineral nature (De La Rocha et al. 2008; Klaas
134 and Archer 2002; Passow 2004). Other studies also suggested that repeated collisions occurring
135 during coagulation processes can lead to more compact particles (Laurenceau-Cornec et al.
136 2015b; Logan and Wilkinson 1990). This mechanism could occur when aggregating particles
137 have a low stickiness and need to collide repeatedly before adhesion occurs. Initially
138 demonstrated for colloids, this mechanism called reaction-limited aggregation or RLA (Ball et al.

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139 1987; Logan and Wilkinson 1990; Meakin 1988) could be extrapolated to larger particles. The
140 opposite mechanism – the diffusion-limited aggregation or DLA (Logan and Wilkinson 1990;
141 Meakin 1988; Witten and Sander 1981) – collate particles with a high stickiness and tend to form
142 loose and porous aggregates.

143 To a lesser extent zooplankton can also influence particle sizes, shape, and structure.

144 Zooplankton feeding activity can fragment large aggregates into smaller particles, the so-called
145 ‘sloppy feeding’ (Dilling and Alldredge 2000; Goldthwait et al. 2004). Zooplankton could also
146 potentially alter the surface roughness and general structure of marine snow aggregates, but to
147 our knowledge there are no reported observations of such an effect.

148 We tested the hypothesis that the presence of minerals and micro-zooplankton feeding can both
149 (in interaction or separately) promote the formation of particles that Stokes’ Law describes well
150 enough so their SV can reliably be predicted theoretically.

151 To test our hypothesis, we experimentally formed phytoplankton aggregates, in the presence and
152 absence of minerals and micro-zooplankton to determine their effect on particle aggregation, and
153 subsequent morphology and SV. Our observed size-SV relationships for the different treatments
154 were then plotted against different parameterization of Stokes’ Law (Eq. 1) and the Navier-
155 Stokes' drag equation (Eq. 3).

156
## Material and methods

### Experimental setting

Two chain-forming diatom cultures — *Skeletonema marinoïi* and *Chaetoceros gracilis* — were aggregated and kept in suspension in cylindrical rolling tanks (Shanks and Edmondson 1989) in the presence and absence of calcite (bio-mineral) and/or rotifers (micro-zooplankton) during 8 days in darkness and at 15°C. A total of twelve 4.5 L cylindrical tanks corresponding to three replicates of four different treatments (P: phytoplankton only; PZ: phytoplankton + micro-zooplankton; PM: phytoplankton + mineral; PMZ: phytoplankton + mineral + micro-zooplankton) were placed on a motorized roller table at t0 and rotated at a speed between 1.5 and 3 rpm. Variable rotating speeds were necessary to keep all particles formed in the different treatments in suspension and to account for SV changes along the experiment duration (see the Results section).

All twelve tanks were filled half with 0.6 µm filtered seawater and half with a mixture of S. marinoïi and C. gracilis grown separately in a modified f/2 medium [Si(OH)₄ = 50 µM; PO₄ = 4 µM; NO₃ = 64 µM; vitamins and metals in f/2 quantities], and diluted to reach final tank concentrations of 65,000 and 110,000 cell L⁻¹ respectively. Calcite was prepared from Cretaceous chalk (see Le Moigne et al. 2013, for further details) and added to the tanks 3, 7 and 11 (PM treatment) and to the tanks 4, 8 and 12 (PMZ treatment), at a final concentration of 1.3 mg L⁻¹. This procedure led to a PIC:POC ratio of 0.1 at the beginning of the experiment, so comparable to but slightly lower than values generally observed for coccolithophore blooms as stated in Le Moigne et al. (2013). Marine rotifers of the genus *Brachionus* sp. were added to the tanks 2, 6 and 10 (PZ treatment) and to the tanks 4, 8 and 12 (PMZ treatment), at an approximate concentration of 6200 individuals L⁻¹ (except tank 10 which received 4400 ind. L⁻¹). This
180 concentration is within the range of typical rotifer concentration encountered in coastal water (up to 8000 ind. L⁻¹; Maioli Castilho and Arcifa, 2000).

182 Aggregate abundance, size and morphology

184 Variations in aggregation and fragmentation processes, abundance, size and morphological
185 properties of aggregates were monitored in the different treatments by imaging the particles
186 present in the tanks at each sampling time. On day 2, 4 and 8, four tanks corresponding to each
187 treatment were removed from the rotating table, placed on their side on a plexiglass plate and
188 aggregates were allowed a few minutes to settle to the bottom. Two cold optic fiber lights
189 (Schott KL 2500 LCD) illuminated the particles from two sides and a 10 megapixels image was
190 made from below through the bottom of the tank with a Canon Powershot G12 camera. All
191 images were analyzed using Fiji, a plugins-enriched distribution of the National Institutes of
192 Health open-source software Image J (Schindelin et al. 2012; Schneider et al. 2012). Pictures
193 were converted to binary (i.e. black and white) using the threshold function. All particles present
194 in the image were counted and analyzed in term of size and shape descriptors (e.g. circularity)
195 provided by Fiji. Following Laurenceau-Cornec et al. (2015a) particle size spectra were obtained
196 by converting particle projected areas to equivalent spherical diameter (ESD), and by binning all
197 particles into 14 logarithmically-spaced size classes (Table 1) to best characterize the whole size
198 range (Jackson 2005; Jackson et al. 1997). Bins containing 5 particles or fewer were excluded
199 from the analysis to ensure statistical relevance (Jackson et al. 2005). The lower size limit was
200 0.87 mm ESD and upper size limit was 129.81 mm ESD. Particle shape and structure were
described by their circularity and boundary fractal number $D_B$ taken as a metrics of surface
complexity (or roughness). $D_B$ also called box-count fractal dimension (or the 'Kolmogorov

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dimension') is the slope of the regression line for the log-log plot of several sets of box sizes (or grids) needed to cover the whole perimeter of the particle and the number of pixels counted inside the boxes for each set of grids (for further explanation on the box-counting method see Kaye et al. 1987; Russ, 2011). $D_B$ was obtained from a box-counting fractal analysis using the Image J plugin FracLac (Karperien 2005). For this fractal analysis a special subset of images consisting of vignettes of a few aggregates that shared the same and best light conditions (i.e. at the intersection of the two light sources) was extracted from the original image. These vignettes were thresholded separately to optimize particle boundary detection and reassembled in collages of aggregates thus offering the best details of their whole perimeter. In addition, only the particles larger than 0.78 mm² in area (equivalent to 1 mm in ESD) were included to give the best results based on the image native resolution. It must be noted that in other sections of the image analysis (e.g. particle size, number and shape), this level of details on particle surface was not required and initial images displaying all particles were used. Bare outline masks images from the Fiji particle analyses were scanned with FracLac by selecting 12 grids per particle (so 12 scans per particle; each scan having a different box size) and letting all other parameters such as maximum and minimum box sizes automatically set by the software.

## Aggregate solid content

In order to infer aggregate densities required for the theoretical SV calculations (using Eq. 1 and 3), the composition of the whole aggregated matter was measured in each tank. After the images were taken, all aggregated matter that could easily be identified (i.e. with a diameter of roughly 1 mm at the minimum) was picked inside each tank and combined in a same beaker using a 10 mL disposable pipette on which the tip had been cut off. For each individual aggregate sampled

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particular care was taken to avoid any loss of material in the process which would have biased the analyses of the relative fractions of their solid constituents. The aggregate mixture was then subsampled (1-3 replicates of 5 to 15 mL) and analyzed for Particulate Organic Carbon (POC) and Nitrogen (PON), biogenic silica (bSiO₂), Particulate Inorganic Carbon (PIC) and Transparent Exopolymer Particles (TEP). TEP being measured following a semi-quantitative method were not used for density calculations but only as a qualitative parameter. Tank water that was cleared from all visible aggregates — hereafter called ‘background water’— was also sampled (1-3 replicates of 10-250 mL) and analyzed to estimate its content in POC, PON, bSiO₂, PIC and TEP. The exact particulate matter concentrations in aggregates were obtained by measuring the particulate content in the small volume of water inevitably sucked into the pipette during aggregate sampling and subtracting it from the aggregate sample concentrations. All protocols followed for POC, PON, bSiO₂ and PIC sample preparation and analysis are described in Le Moigne et al. (2013). TEP were measured using the spectrophotometric method from Passow and Alldredge (1995). A 0.02 % aqueous solution of Alcian Blue in 0.06 % acetic acid was used to stain the TEP contained in samples. In each tank three replicates of 10 mL of background water and 10 mL of the aggregate mixture were sampled and filtered onto 0.4 µm Whatman polycarbonate filters at a constant vacuum below 200 mbar to prevent the TEP from being pulled through the filter pores. The filters were then stained with 500 µL of 0.02 % Alcian blue solution and soaked for 2h in 6 mL of 80 % H₂SO₄ to dissolve the stain. The absorbance of the solution was measured at 787 nm with a PRIM Light and Advanced Spectrophotometer. The calibration curve relating mass of xanthan gum to absorbance was established using a protocol modified from Passow and Alldredge (1995) to avoid problems potentially linked to a heterogeneous xanthan gum solution. Instead, 0.4 µm polycarbonate filters were disposed onto a
Mettler Toledo scale with a sensitivity of 1 µg, tared, and 40 to 800 µg of xanthan gum powder (VWR, USP/NF) were deposed directly onto the filters. These filters were then stained with 500 µL of 0.02 % Alcian blue solution at a constant vacuum under 200 mbar and soaked for 2 h in 6 mL of 80 % H₂SO₄. Finally, absorbance of the solution was measured at 787 nm.

## Aggregate volume, porosity and density

Aggregate density ($ρ_{agg}$) was calculated in each tank from the volume fractions in the solid content of POM (taken as the sum of POC and PON), PIC and bSiO₂ in the aggregate mixture — used to calculate the density of the aggregate solid content ($ρ_{sol}$) — and from the aggregate porosity (i.e. fraction of the aggregate volume occupied by water) using the following formula:

$$ρ_{agg.} = ρ_{sol.}×(1 - P) + ρ_{sw}×P \quad (4)$$

where $ρ_{sw}$ is the density (kg m⁻³) of the seawater contained in the aggregate and P is the aggregate porosity. Densities for POM, PIC and bSiO₂ were respectively set at 1060, 2710 and 2000 kg m⁻³. Porosity was calculated using,

$$P = 1 - \left( \frac{Vol._{POM+PIC+bSiO_2}}{Vol._{agg.}} \right) \quad (5)$$

where Vol.ₚOM+PIC+bSiO₂ is the total volume of the solid content in aggregates and Vol.ₐgg. is the total volume of aggregates estimated from image analysis. Only aggregates larger than 1 mm ESD were included to keep only those that were manually picked for the chemical analyses. Individual aggregate volume was calculated from the minor (min. ax.) and major axes (maj. ax.) of the best fitting ellipse on particle images given by Fiji and assuming a prolate shape, using the following formula:

$$Vol._{agg.} = \frac{4}{3} \times \pi \times \left( \frac{min. ax.}{2} \right)^2 \times \frac{maj. ax.}{2} \quad (6)$$
268 Aggregate excess densities ($\Delta\rho$) were calculated assuming a seawater density of 1025.3 kg m⁻³
269 (corresponding to 15 °C and a salinity of 34).

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271 Sinking velocity measurements
272 *Observations*
273 Aggregate SV were measured in vitro over the course of the experiment using the non-invasive
274 video recording technique developed by Ploug et al. (2010). This method permits continuous
275 measurements during incubation. It allowed to monitor the SV in each treatment at a higher
276 temporal resolution (day 2, 3, 4, 5, and 8; Table S1, supplementary material) than other
277 measurements requiring to end the incubation. Calculations are based on incremental x- and y-
278 positions of particle orbital trajectories in relation to the center of the tank side being taken as a
279 spatial reference. Measurements aimed to cover the largest size range offered by the particle
280 population inside the tank but particle choice was also constrained by their adequate position and
281 orbits required for post-recording analyses. A cold optic fiber light from a Schott KL 2500 LCD
282 was used to illuminate the rotating particles without disturbing their orbital trajectories and video
283 recording was conducted with the same Canon Powershot G12 camera used for particle imaging.
284 The method from Ploug et al. (2010) was completed here by the use of a laser-pointer to identify
285 in the video sequences each individual aggregate being filmed and record its distance from the
286 side wall of the tank along an axis parallel to the rotation axis (z-axis). In the video footage, the
287 apparent x, y coordinates and size of the aggregates are affected by its distance from the side
288 wall of the cylinder through which the video is shot due to the deformation of perspective. Each
289 x and y-positions of the aggregates and their size were corrected from this distortion effect using
290 their distances from the side wall recorded for each particle. A total of 203 paired measurements
291 of aggregate size and SV was conducted.

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293 *Theory and models*
294 Theoretical aggregate SV were obtained using Stokes' Law (Eq. 1) and the Navier-Stokes
295 equation which includes the drag coefficient in its formulation (Eq. 3). Aggregate excess
296 densities (Δρ) required in both equations were calculated using two different parameterizations
297 for aggregate porosities and introduced in each formulation, leading to four different models that
298 were compared to observations.

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Here, a constant density was assumed for all aggregates having the same origin (from the same
treatment and same tank). Aggregate densities were inferred from the relative fractions of their
solid components in POM, bSiO₂, PIC measured on the aggregate mixture (see section
'Aggregate volume, porosity and density'). Porosities were fixed at 0 and 99% to mimic solid
spheres and highly porous particles close to values reported in the literature for large marine
aggregates such as marine snow (Alldredge and Gotschalk 1988).

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*Model 2: Navier-Stokes drag equation (Eq. 3) with a constant aggregate density*

In model 2, aggregate densities were calculated from the measured solid content (similar to model 1) and a constant porosity of 99%. Empirical drag coefficients were fixed at values of 1 and 5.

310 *Model 3: Stokes' Law (Eq. 1) with aggregate density scaled on a fractal geometry*

<figure><img src="image_17.png" /></figure>
Model 3 used a constant density for aggregate solid contents from measurements but assumed a porosity increasing with increasing size based on a fractal scaling using the formula proposed by Logan and Wilkinson (1990) and following Laurenceau-Cornec et al. (2015b):

$$P = 1 - a \times l^{D_3 - 3} \qquad (7)$$

where P is aggregate porosity, l is the largest diameter, a is an empirical constant and $D_3$ is the three-dimensional fractal number. Figure 1a shows the size-porosity relationship for different values of a and $D_3$. Note the constant porosities (horizontal lines) at all sizes obtained for $D_3 = 3$ that corresponds to a Euclidean geometry for which particle volume is filled at the same rate than particle size increases. In this case $P = 1 - a$, and the porosity depends only on the value attributed to a. For small values of a and large $D_3$, the rate at which porosity increases with size is maximum at the smallest sizes. For model 3, $D_3$ was fixed at 1.4 and 1.8 following values obtained in the literature and describing a large range of marine aggregate structures (Logan and Kilps 1995). The constant a was fixed at 0.03 following Logan and Wilkinson (1990).

### Model 4: Navier-Stokes drag equation (Eq. 3) with aggregate density scaled on a fractal geometry

Model 4 infers both the porosity and drag coefficient assuming a dependence on size based on a fractal scaling. Variation of the drag coefficient with size for various values of $D_3$ and a is shown in Figure 1b. The fractal porosity was calculated as described above with $D_3$ values of 1.4 and 1.8 and a = 0.03. Calculations of the fractal drag coefficient followed Laurenceau-Cornec et al. (2015b). Eq. 3 was solved for the drag coefficient using two paired values of size and SV defining specific drag conditions [called ‘specific conditions’ in Laurenceau-Cornec et al. (2015b)]. A bisection algorithm (cost function; code available in supplementary material) computed on Matlab (Simulink) allowed to solve the White function numerically (White 1974)

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and to calculate the drag coefficients at each size from Reynolds numbers using Stokes' assumption as a starting point ($Re_0 = 24 / C_{D0}$). The White function is one of the numerous different empirical formulations for the drag coefficient (Clift and Gauvin 1971; Kelbaliyev 2011) and was chosen here for its simple expression. Aggregate solid fractions needed in the calculations of Figure 1b were arbitrarily set at 60% POM and 40% bSiO$_2$; these values correspond roughly to the measurements obtained for the P treatment on day 9 (see Results section). For large values of $a$, Stokes' Law and the White function give similar $C_D$ values up to very large sizes (e.g. specific drag condition 1, $a = 0.1$) suggesting that Stokes' Law could be valid in these conditions. Note the poor control of $D_3$ on $C_D$ variations except at very small sizes and coefficient $a$.

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## Error assessment and statistical analysis

The quadratic propagation of error from the potential erosion or dilatation of particle projected areas during image thresholding (see section 'Aggregate abundance, size and morphology') to particle porosity and density calculations was assessed using rules derived from the Gaussian equation for normally-distributed errors. Particles that presented most of the alteration after threshold were the largest and most complexly-shaped. For these particles a homogeneous illumination of the whole surface was difficult. A filter fixed at 3 mm$^2$ projected area (equivalent to a 2 mm ESD) and 0.5 circularity was applied to each tank binary image to keep only the particles suspected to have generated most of the error. Then the effect of eroding or dilating these particles on the total aggregate volume was assessed for each tank. Our results show that a removal of a layer of 1 pixel from these particle contours (erosion) induces an average total aggregate volume loss of $7.2 \pm 3.8 \%$, while adding a layer of 1 pixel (dilatation) induces an

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average total volume gain of 10 ± 4.5 %. Based on this, calculations of propagated errors on porosity and density assumed a ± 10 % variation of particle volume due to image binary conversion. Standard deviations of the replicate measurements of POC, PON, bSiO2 and PIC contents in aggregates were also used in the calculations of error propagations.

One-way analyses of variance (ANOVA) were conducted using the language and environment for statistical computing R to test the statistical significance of the differences in the mean ESD, particle boundary fractal numbers and circularities between treatments. Homogeneity of variances were tested using a Bartlett's test, and non-parametric Kruskal-Wallis rank sum tests were also used as means to confirm treatment differences. All pairwise comparisons between treatments were performed following the Tukey's method (differences in mean levels) to identify difference between treatments.

Non-linear (power) regression fits to the size-SV relationships for this study and others shown here were also performed with R. Along with the regression coefficient, standard errors, p-values and profile likelihood confidence intervals (95 %) were derived from the model fits and are provided as possible metrics to evaluate the goodness of fits.

To assess the performance of the four model parameterizations to predict experimental data, the Residual Sum of Squares (RSS) and Residual Mean Square Error (RMSE) between predicted and observed values were calculated. Smallest values of RSS and RMSE were considered to indicate the best model fit to observations. Additionally, the difference in logarithms of the likelihood Δlog(L), based on a Gaussian error distribution with a constant variance of log-transformed data, was taken as a criterion to inter-compare model performance (Armstrong et al. 2002, and references therein). The log(L) was calculated as in Moriceau et al. (2009):

$$ \log(L) = -\frac{N}{2} \log \left( \frac{\sum (\log(\hat{SV}_i) - \log(SV_i))}{N} \right) \quad (8) $$
378 where *N* is the number of data points, *SV*<sub>*i*</sub> is the measured SV at data point *i*, and *SV̂*<sub>*i*</sub> is the corresponding value predicted by the model. After pairwise model comparisons, a difference in log(*L*) of at least two points was considered as a criterion of a better fit to the observations (Hilborn and Mangel 1997).

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384 Results

385 Aggregation-disaggregation processes

386 The presence or absence of calcite and micro-zooplankton induced significant variations in
387 particle size, shape and general aspect (Fig. 2). Direct observations made on day 4 (Fig. 2a to d)
388 suggest that the addition of calcite and micro-zooplankton (PZ, PM and PMZ) promoted the
389 formation of smaller aggregates than with phytoplankton alone (P). Aggregates in the P and PZ
390 treatments appeared more ‘fluffy’ (i.e. porous) and less round than in the PM and PMZ treatment,
391 suggesting that calcite could induce an increase of particle compactness and sphericity, an effect
392 not observed in the presence of micro-zooplankton alone. By day 8 (Fig. 2e to h), most of the
393 differences between treatments previously visible by direct observations diminished significantly.
394 Small tails appeared at particle surfaces in the PM and PMZ treatments (Fig. 2g and h, white
395 arrows) as it is often observed on fast-sinking particles (see for example Fig. 1 in Engel et al.
396 2009).

397 Aggregates formed between days 0 and 4 in the following chronological order: PM and PMZ
398 first, then PZ and finally P being the last to form. The maximum number and volume of > 1 mm-
399 sized aggregates were reached on day 4 in the P and PZ treatments, while it peaked on day 2 in
400 PM and PMZ treatments (Table 2) suggesting that faster aggregation occurred in the presence of
401 calcite. After the maxima in aggregate number and volume a decreasing trend was observed in
402 all treatments. Monitoring of particle size spectra (Fig. 3) confirmed that most of the aggregation
403 processes took place in all treatments from day 2 to 4 (no data for the P treatment for which no
404 visible aggregates were formed on day 2). This result was revealed by a transfer of particles from
405 small size classes to larger ones. In the PZ, PM and PMZ treatments, the number of aggregates in
406 size classes ranging from 1 to 3 mm ESD approximately varied the least during aggregation
processes, suggesting that this particular size range represented the most stable size reached in these treatments. An increase of the smallest particles in PM tanks between days 2 and 4 showed that disaggregation processes also took place in this treatment during this time period. Days 4 to 8 were mostly marked by disaggregation processes illustrated by less of the large particles combined with more of the small ones. Particles sized around 1-2 mm were again those that participated the least to disaggregation processes in the PM and PMZ treatments confirming the equilibrium status of this size range. Furthermore, in the PM treatment the size class 8 (1.1 - 1.5 mm) was the only class that showed an increase in particle number, while both smaller and larger particles displayed losses. The size class of particle that varied the least between days 4 and 8 in both P and PZ treatments were smaller around 0.4 - 0.5 mm.

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## 418 Aggregate characteristics

### 419 Aggregate size, circularity and boundary fractal number

During the experiment the largest mean ESDs (and also largest standard deviation) was observed on day 4 in the P treatment with a value of 2.53 ± 1.28 mm (Fig. 4a; Table 3), suggesting that these aggregates were the largest but also the least uniform in size. On day 8, aggregates from the P treatment had the smallest mean and median ESDs. All other aggregates (PZ, PM and PMZ treatments) had more uniform sizes and displayed little variations during the experiment, confirming results from size spectra variations (see above). On day 4, statistical tests (ANOVA, Kruskal-Wallis test and Tukey pairwise multiple comparison of means) confirmed a significant difference between the mean ESDs of P aggregates and those from other treatments except with PMZ (Table 3; Tukey *p*-value = 0.217 for the pair PMZ-P). No statistical difference could be

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429 found between the mean ESDs of all treatments on day 8, confirming direct observations that
430 aggregates became more similar toward the end of the experiment.

431 Analyses of aggregate circularities showed that particles formed in the presence of calcite and
432 micro-zooplankton (PZ, PM, PMZ) were the most circular (Fig. 4b and Table 3). Conversely, P
433 aggregates had the lowest median circularities, agreeing with results suggesting more complex
434 shapes for these aggregates (see above). ANOVAs and Kruskal-Wallis tests confirmed the
435 statistical differences of circularity between treatments (Table 3). Pairwise multiple comparison
436 of means (Tukey contrasts) showed that P aggregates were significantly different in circularity
437 from other treatments except on day 8 for which no statistical difference could be made between
438 P and PM aggregates.

439 Aggregate boundary fractal number from image analyses mirrored the circularity results and
440 suggested that aggregates formed in the presence of calcite and micro-zooplankton had less
441 complex surfaces (i.e. were smoother) than if produced from phytoplankton alone (Fig. 4c). For
442 each sampling day, box-counting analyses showed that aggregates formed in the presence of
443 calcite (PM and PMZ treatments) had the lowest median boundary fractal number D<sub>B</sub> (Fig. 4c
444 and Table 3), except on day 8 for which the PZ treatment showed the lowest mean D<sub>B</sub>. ANOVAs
445 and Kruskal-Wallis tests confirmed the statistical significance of the differences in mean D<sub>B</sub>
446 between treatments (Table 3). Pairwise multiple comparisons of means (Tukey contrasts) showed
447 in particular that P treatments significantly differed from others except for the pairs PZ-P and
448 PM-P for which no statistical difference could be established on day 4 and 8 respectively (Fig. 4c
449 and Table 3). The largest variation of median D<sub>B</sub> was observed for the PZ treatment between
450 days 4 and 8 with a 0.034 decrease suggesting a significant surface smoothing of the particles.

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452 Aggregate composition and porosity

453 An increase of all total particulate matter (POM, PIC and bSiO₂) content was observed between
454 days 2 and 4 in all treatments— so during on-going aggregation processes— while a decrease of
455 POM concentration occurred between days 4 and 8 suggesting degradation processes (Table 2).
456 As expected, relative mass fractions of POM, PIC and bSiO₂ in aggregate solid contents (see
457 Table S2 in supplementary material for volume fractions) showed large differences between
458 treatments, especially those that were grown in the presence of calcite or not. Dry content of
459 aggregates from the PM and PMZ treatments contained in average 54.8 ± 8.4 and 51.2 ± 7.9 %
460 of PIC respectively, while those from the P and PZ treatments had 4.4 ± 2.4 and 4.1 ± 1 % of
461 PIC, confirming that calcite was important for aggregation in the PM and PMZ treatments. POM
462 represented the largest fraction of the solid content in P and PZ treatments (average values 69.9 ±
463 15.9 % and 69.7 ± 7.1 % POM respectively). Decreases of the fraction of organic components
464 (POM) between days 4 and 8 were accompanied by large increases of bSiO₂ toward the end of
465 the experiment (increases of 25.9 ± 2.3, 4.2 ± 1.7, 4.1 ± 1.3 and 6.2 ± 0.8 % bSiO₂ in P, PZ, PM
466 and PMZ treatments respectively). Between days 2 and 8, the PIC fraction decreased
467 continuously in the P and PZ treatments (decreases of 3.4 ± 0.9 and 1.9 ± 0.4 respectively) while
468 it increased in the PM and PMZ treatments (increases of 16.4 ± 10.9 and 14.1 ± 2.5 respectively).
469 Total TEP content in aggregates increased between days 2 and 4, and decreased toward day 8 in
470 all treatments, except in the P treatment where TEP increased between days 4 and 8 (Table 2).
471 Aggregates from the P treatments had the highest porosities (average of 99.76 ± 0.1 % of
472 seawater). The lowest porosities were observed for the PM and PMZ treatments on day 8 (99.33
473 ± 0.11 and 99.32 ± 0.11 % of seawater respectively), suggesting that these particles had the most
474 compact structure.
As expected, aggregates that incorporated calcite had the highest excess densities illustrating the ballast effect (averages of 3.73 ± 1.7 and 3.57 ± 1.5 kg m⁻³ in the PM and PMZ treatments for the whole duration of the experiment), while aggregates from the P and PZ treatments had the lowest excess densities (averages of 0.6 ± 0.42 and 1.27 ± 0.38 kg m⁻³ in the P and PZ treatments respectively).

Observed sinking velocities and model predictions

Measured SV presented very different ranges between treatments that contained calcite and those that did not (Fig. 5 and Table 4). Ballasted aggregates (PM and PMZ treatments grouped) had an average SV of 1045 ± 310 m d⁻¹ (min = 327 m d⁻¹; max = 1704 m d⁻¹) while aggregates without added calcite (P and PZ treatments grouped) had an average SV of 456 ± 136 m d⁻¹ (min = 218 m d⁻¹; max = 928 m d⁻¹). This result confirms the expected ballast effect induced by the addition of calcite. The goodness of the fits to the relationship between ESD and SV varied as a function of sampling day and treatment. Overall, power law regressions presented a better fit to the relationship between ESD and SV for aggregates formed in the PM and PMZ treatments with an improvement over time towards the end of the experiment (Table 4; lowest standard errors and *p*-values < 0.001). Conversely, higher standard errors and *p*-values on coefficient estimates were obtained when fitting power law regressions to the data of the P and PZ treatments. This result suggests that size is a better predictor of the SV for ballasted aggregates than for those grown with phytoplankton only. Conversely, the goodness of the fit to the size-SV relationship was not improved by the addition of zooplankton (comparisons P vs PZ and PM vs PMZ), suggesting that, unlike calcite, zooplankton do not increase the control of size on aggregate sinking velocities.
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Fits of the four models using Stokes' Law and the Navier-Stokes drag equation (see section 'Theory and models') to the observations are reported in Figure 5. Quantitative estimation of model performances using the Residual Sum of Squares (RSS) and the Root Mean Square Errors (RMSE) showed that model 4 (Navier-Stokes drag equation parameterized with an aggregate density scaled to fractal geometry; $D_3 = 1.4$; $a = 0.03$; Fig. 5d) presented the best fit to the P and PZ data (Table5; average RSS of $0.16 \pm 0.15$ and average RMSE of $0.1 \pm 0.03$). Also, model 3 (Stokes' Law parameterized with an aggregate density scaled to fractal geometry; $D_3 = 1.8$; $a = 0.03$; Fig. 5c) best reproduced PM and PMZ data (Table 5; average RSS of $0.43 \pm 0.51$ and average RMSE of $0.11 \pm 0.04$). The criterion of a minimum two-points higher difference in $log(L)$ between models 2 and 4 for the P and PZ treatments confirmed better performance for model 4 in tanks 9 and 2 [$\Delta log(L)$ of 4.29 and 2.18 respectively]. In the PM and PMZ treatments, model 3 presented a higher $log(L)$ of at least 2 points than model 1 in tanks 3, 7 and 4 [$\Delta log(L)$ of 2.97, 8.9 and 2.75 respectively], while model 1 performed better than model 3 only in tank 11 [$\Delta log(L)$ of 3.92] where the greatest SV were observed confirming the RSS and RMSE results.

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514 Discussion

515 Uncertainties from sampling and temporal resolution

516 Aggregates that were sampled and grouped in a mixture at the end of each tank incubation had a
517 minimum diameter of ca. 1 mm (see section ‘Aggregate solid content’). Fractions of POM,
518 bSiO₂ and PIC were measured in these mixtures and used to calculate general densities for the
519 aggregated material formed in each tank. Matter smaller than 1 mm was considered unidentified
520 and intentionally excluded from our sampling to avoid any possible contamination of our
521 mixtures. Despite this precaution, uncertainty lies in potential variations of the solid component
522 fractions across the population of individual aggregates that composed each mixture. This risk is
523 high for aggregates formed from natural phytoplankton communities (Alldredge and Silver 1988;
524 Laurenceau-Cornec et al. 2015b) but we believe that our aggregates formed in controlled
525 conditions from uniform phytoplankton cultures presented very similar composition. The limited
526 scattering of the data points originating from similar treatments (Fig. 5) combined with the
527 ability of our models using these measured compositions to reproduce fairly well the
528 observations gives us good confidence that these variations were limited.

529 Another uncertainty pertains to the temporal resolution of our sampling. Because of logistic
530 constrains related to our numerous measurements, sampling times were restricted to day 2, 4, and
531 8, with two additional times at day 3 and 5 for the SV measurements only. It implies that
532 fluctuations in aggregate physico-chemical properties during ongoing aggregation/disaggregation
533 or remineralisation could have occurred between two consecutive sampling points without being
534 resolved by our measurements. Our results mainly reflect the general trends between two time
535 steps and can identify the transient states and especially the tipping points at an accuracy scaling
536 between one and four days. For instance, the onset of degradation as possibly suggested by a
transition from TEP increase to decrease at day 4 (Table 2; exception for P treatment) might have occurred earlier (between day 2 and 4) or later (between day 4 and 8) and at a different peak value. Results and conclusions drawn here are however not highly sensitive to this uncertainty since we mostly relate measurements of aggregate physico-chemical properties to their sinking velocity measured at near or similar sampling times with little focus on the effect of transient states.

Minerals and zooplankton roles in Stokes' Law applicability

Our initial hypothesis was that incorporation of minerals and zooplankton feeding on marine aggregates induce modifications of particle properties that allow a reliable use of Stokes' Law. Results from our experiment confirmed that particles formed in the presence of calcite (PM and PMZ treatments) have a size-SV relationship that consistently follows Stokes' models. In particular, the overall best fit to the PM and PMZ size-SV data was obtained for model 3 (parameterization of Stokes' Law with a porosity scaled to a fractal geometry, Fig. 5c).

Interestingly, model 1 (parameterization of Stokes' Law with a constant porosity) showed better prediction than model 3 for the fastest-sinking aggregates that were formed at the end of the incubation (tanks 11 and 12 on day 8). This result suggests that aggregates from tanks 11 and 12 were the best candidates to fit a traditional Stokes' Law model. These aggregates would have acquired properties that, among all particles formed in this experiment, were the closest to those of a 'Stokes' particle' (i.e. a small perfectly smooth sphere settling at low Reynolds number). Average ESDs of aggregates from tanks 11 and 12 were the largest among their respective treatments (PM and PMZ), thus not supporting a small size as a possible explanation for their better fit to a traditional Stokes' curve. However, their low porosities (Table 2) suggest that these

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particles were very compact and likely had a structure better represented by a Euclidean geometry than a fractal geometry. This observation tends to confirm the suspected effect of minerals on decreasing particle stickiness and enhancing aggregate compactness through Reaction-Limited aggregation. The fractal nature of marine aggregates implies that small ones are expected to be more compact than large ones due to the increase of porosity with size as already observed experimentally by Engel et al. (2009). By leading to a loss in stickiness, the presence of minerals could counteract to some extent the fractal building of marine aggregates by forcing each building block to more efficiently fill the volume available and follow to a higher extent a Euclidean geometry. One would then expect mineral-loaded aggregates to have a $D_3$ fractal number higher than those usually observed for fractal marine aggregates. Logan and Wilkinson (1990) compared the fractal numbers of various marine aggregates formed in different environments. They reported fractal numbers for marine snow of 1.26 (Alldredge and Gotschalk 1988) while aggregates collected in an estuary – thus having a higher terrigenous mineral fraction – had a higher fractal number of 1.78 (Gibbs 1985). Boundary fractal numbers $D_B$ reported here are however not comparable with values of one-, two-, or three-dimensional fractal numbers (noted $D_1$, $D_2$, and $D_3$ respectively and themselves inter-comparable only in specific conditions) as they are similar in meaning but use different calculation methods (Kilps et al. 1994; Russ 2011). However, low values of $D_B$ as observed especially for tanks 11 and 12 on day 8 tend to indicate a low surface complexity as expected for more compact and smooth particles. Particles formed in the presence of micro-zooplankton, but without calcite (treatment PZ), did not show marked differences from the P treatment, suggesting a limited effect from micro-zooplankton presence alone. This result does not confirm our initial hypothesis that micro-zooplankton could alter the size-SV relationship in a similar manner to calcite. PZ aggregates on

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583 days 2 and 4 showed significantly smaller circularities and higher $D_B$ values than for the PM and
584 PMZ treatments but had on day 8 all properties allowing to expect a much better fit to a Stokes'
585 Law model. Only excess density was significantly larger for the calcite treatment compared to
586 the micro-zooplankton treatment on day 8 due to the ballasting effect of calcite. This result is
587 nevertheless counter-intuitive as Stokes' Law is deemed valid at $Re < 0.5$. This means that at
588 constant seawater kinematic viscosity $\nu$ (see Eq. 2), Stokes' Law should be valid for small and
589 slowly-sinking particles. Regardless of the treatment, none of the particles formed in this study
590 had a $Re$ value lower than 0.5 (range between 2 and 50; Table 4), and our results suggest that the
591 fastest-sinking particles, those ballasted with calcite, were those that agreed best with Stokes'
592 models. Several previous studies of natural and artificially-formed marine snow reported values
593 of Reynolds number comparable to ours (Alldredge and Gotschalk 1988; Engel et al. 2009). This
594 is precisely the range in which $C_D$ starts to differ significantly between Stokes' Law and other
595 empirical approximations like Whites' (see for example Fig. 1 in Loth 2008). The $Re < 0.5$ limit
596 for the validity of Stokes' Law is empirical and based on fluid flow observations past spherical
597 sinking particles (Loth 2008; White 1974). Our results suggest that for $1 < Re < 50$, Stokes'
598 regime may remain reasonably applicable for phytoplankton aggregates given that they fulfil
599 certain criteria in terms of roundness, roughness and compactness that could possibly be acquired
600 in case of mineral incorporation. This result provides insights as to why the PZ aggregates —
601 satisfying these criteria and having Reynolds numbers lower than those observed for the particles
602 that agreed best to the Stokes' model (tanks 11 and 12) —, did not fit better to Stokes' models.
603 Different drag properties that were not adequately captured by our boundary fractal number
604 analysis could explain this discrepancy. Maggi (2015) showed experimentally that internal and
605 external fractal characteristics of the aggregates can induce non-linear hydrodynamic effects
through their influence on surface roughness and porosity. In particular, feeding activity of the rotifers at the surface of the aggregates could have led to an increase of surface roughness at a scale unexpectedly not resolvable on our images given rotifers' small size (usually a few hundred microns) compared to our millimeter-sized aggregates. In addition, rotifers used in this experiment are micro-zooplankton and cannot serve as a model for all zooplankton. It is likely that larger species would have interacted differently with aggregates and probably led to contrasting results.

Conversely, the best fits for all non-calcite ballasted particles (P and PZ treatments) were obtained with the Navier-Stokes drag equation. Model 4 parameterized with a fractal porosity and drag performed well. This result confirmed our expectations that aggregates grown in absence of calcite should have characteristics farther from a theoretical Stokes' particle because of their more irregular shape, higher porosity and more complex fractal structure. It is also in agreement with the numerous studies that documented the poor control of size on SV for large, loose and highly-porous phytoplankton aggregates. This can be partly explained by the increase of porosity with size for fractal aggregates that can even lead to a decrease of the SV with size for the largest particles (Laurenceau-Cornec et al. 2015b).

## Relevance to other ballasted and non-ballasted aggregates

To test whether our findings remained valid with a larger dataset, we gathered size-SV data from other numerous experimental and field studies (Alldredge and Gotschalk 1988; Alldredge and Gotschalk 1989; Azetsu-Scott and Johnson 1992; Carder et al. 1982; Diercks and Asper 1997; Engel et al. 2009; Gibbs 1985; Iversen et al. 2010; Iversen and Ploug 2010; Iversen and Robert 2015; Jouandet et al. 2011; Kajihara 1971; Laurenceau-Cornec et al. 2015b; Nowald et al. 2009;

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629 Ploug et al. 1999; Shanks and Trent 1980; Syvitski et al. 1995). We refer the reader to

630 Laurenceau-Cornec et al. (2015b) if more information on these studies (particle type, method

631 used, location) are needed. We split data into two categories – ballasted and non-ballasted

632 aggregates – and fitted on these two groups parameterizations of models 3 and 4 (Fig. 6). It is

633 important to note that this classification is arbitrary and that drawing a clear limit between these

634 two types of aggregates cannot represent the reality as it exists a continuum of mineral loads in

635 aggregates — from none (i.e. phytoplankton that do not produce mineralized skeletons) to those

636 that include minerals produced naturally by phytoplankton, and exogenous minerals scavenged

637 in the water column. Empirically, we considered all aggregates without exogenous mineral as

638 non-ballasted (note that our P and PZ treatments contained biogenic silica from the diatom

639 frustules) and all aggregates that included exogenous minerals as ballasted (one exception noted

640 on the figure caption). For non-ballasted aggregates, Figure 6a shows that the use of model 4

641 (Navier-Stokes drag equation) parameterized with various fractal numbers, initial conditions and

642 coefficient *a* fitted the measured size-specific SV best. Again, this confirms the known

643 complexity of predicting the SV of phytoplankton aggregates from their size and density alone. It

644 also highlights the crucial importance of the fractal structure and drag characteristics in the

645 control of the SV for these aggregates and that no reliable prediction of the SV could be achieved

646 without considering these particle characteristics. Figure 6b shows various datasets of aggregates

647 that incorporated minerals either in situ or artificially in experimental conditions. With the

648 exception of a set of aggregates, most of the size-SV relationships display a similar slope. Our

649 model 3, a Stokes’ Law model parameterized with a fractal porosity, reproduces well this general

650 slope with a fractal number of approximately 1.8. The different SV offsets needed to fit the

651 model to each dataset are easily obtained by varying the coefficient *a* between 0.1 and 0.001.

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This result strongly supports our findings and confirms that a Stokes' parameterization can reliably be used to predict the SV of aggregates including minerals. It also highlights the already noted importance of the dimensionless empirical constant *a* in this parameterization (Laurenceau-Cornec et al. 2015b) and poses the question of how to constrain it. Equation 6 suggests that at a given *D*<sub>3</sub> and particle size, the porosity is directly proportional to *a*. This indicates that assuming a known *D*<sub>3</sub> — set for instance at 1.8 based on Figure 6b — a prediction of the SV of ballasted aggregates as a function of their size and density would be possible using Stokes' Law but only given data on the value of *a*. At similar aggregate sizes and densities, the SV offset between the various sets of aggregates would thus be created only by slight differences in porosity, with the highest SV corresponding to the lowest porosities and vice versa.

We investigated the extent to which the concentration of minerals in the aggregates could be important in determining the acquisition of Stokes' like properties by the particles. In all studies reported in Figure 6b, aggregates carried between ~7 and 98% of various minerals (mineral:dry weight ratio), which is a very large range of mineral load. This gives us confidence that the mechanism identified here could be highly relevant to all mineral-loaded aggregates possibly regardless of mineral nature. According to Le Moigne et al. (2014), surface POC flux associated with minerals could represent 60 % of the POC flux in the Atlantic and 40 % in the Southern Ocean and Le Moigne et al. (2012) estimated over 55 locations in the North Atlantic and Southern Ocean that approximately 32 % of the sinking POC flux was associated with minerals. Atmospheric dust deposition that brings lithogenic particles to the surface ocean are important events that increase the proportion of mineral-associated POC in the surface ocean (Bressac et al. 2014; Guieu et al. 2014; Pabortsava et al. 2017). Interestingly, the increase of the POC:mineral ratio in aggregates in relation to ocean acidification has been shown to significantly alter the

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675 size-SV relationship with clear deviations from Stokes' Law (Biermann and Engel 2010). It is
676 also important to consider how the proportion of POC:mineral changes with depth. The
677 preferential remineralization of POC through the upper mesopelagic results in an increase of the
678 mineral fraction inside the aggregates (Honjo et al. 1982; Iversen and Ploug 2010). Mineral
679 dissolution which also takes place with depth has been shown to occur at limited rates in large
680 marine aggregates mainly due to their size, high sinking velocity and the micro-environment that
681 they form (Jansen et al. 2002; Moriceau et al. 2007). Applying a Stokes' parameterization to the
682 particles sinking through the mesopelagic and deeper could be more relevant than in the surface
683 ocean. The acquisition of Stokes' like properties at depth through a loss of POM could also apply
684 to aggregates that are ballasted only through their load of mineralizing phytoplankton cells. This
685 is supported by several field studies that documented an increase of aggregate sinking speed with
686 depth (Berelson 2002; Trull et al. 2008; Villa-Alfageme et al. 2016) and biogeochemical models
687 include parameterizations of SV increase with depth to match the observed POM concentration
688 profiles (Aumont et al. 2015; Gehlen et al. 2006; Kriest and Oschlies 2008). This is to some
689 extent what our experiment reproduced. The drastic decrease of the fraction of POM in
690 aggregates in all treatments testifies to the degradation that occurred in the tanks over the course
691 of the 8 days of incubation (Le Moigne et al. 2013). It is also indicated by the decrease in TEP
692 content observed in all tanks (except for the P treatment) between days 4 and 8. Moreover, we
693 recorded a continuous increase of aggregate SV during the experiment (Table 4), suggesting a
694 possible increase in excess density due to changing ratios of low (organic) to high (mineral)
695 density components. However, even towards the end of the experiment, aggregates from the P
696 treatment still did not present Stokes' like properties and other studies reported that an organic
697 matter loss at depth was accompanied by an increase of aggregate porosity (McCave and Gross
698 1991) and reported particularly low SV not correlated with size (Diercks and Asper 1997). In this
case, the empty space created by a loss of organic matter by remineralization would be replaced
699
by water — with a preservation of the initial architecture of the aggregate — instead of a
700 rearrangement that would possibly increase aggregate compactness thus promoting the
701 acquisition of Stokes' like properties. The increase in TEP content between days 4 and 8
702 observed only in the P treatment could also suggest a substantial bacterial growth (Decho 1990)
703 leading to the production of low-density matter. Unfortunately, there are no available in situ data
704 on porosity and size-SV relationship for individual particles in the deep ocean (> 1000 m) to
705 draw any firm conclusions.
706

707 We identified how the addition of calcite and micro-zooplankton can influence particle aspect,
708 size, shape and structure. Based on our results, calcite and to a lesser extent micro-zooplankton
709 grazing are responsible for a significant increase in particle roundness and compactness, which
710 improve Stokes' predictions. As expected, the inclusion of calcite in our particles significantly
711 increased their SV, which in turn significantly modified their shape, especially for the fastest-
712 sinking aggregates. In fact, circularities of particles in tanks 11 and 12 (those for which the best
713 fitting to Stokes' Law was obtained) were unexpectedly low if compared to the PZ treatment.
714 Small tails were observed on these particles (Fig. 2g and h) and we attribute the loss in
715 circularity to these small changes in particle shape. This result further suggests that roundness is
716 necessary but not sufficient in the acquisition of Stokes' properties but that the level of
717 compactness is likely to be the most important factor to consider when trying to predict the SV
718 of marine aggregates with a Stokes' model.

719 In addition to its effect on particle physico-chemical properties, calcite significantly influenced
720 aggregation and disaggregation dynamics. In particular, the most stable size range (or ‘the

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equilibrium size') at which the transfers between small to large (during coagulation) or large to small (during fragmentation), was higher in the treatments that contained calcite. In other terms, phyto-aggregates that did not include calcite, fragmented into smaller aggregates than did those that included calcite. This is an interesting result as it contradicts the common view that minerals tend to induce the formation of smaller aggregates essentially via fragmentation processes (Engel et al. 2009; Hamm 2002; Iversen and Ploug 2010; Passow and De La Rocha 2006). Moreover, in the PM treatment between days 4 and 8, both small and large particles converged to one single size class (~1-1.5 mm) via aggregation and fragmentation processes. It is likely that the fragmentation process for the mineral-loaded aggregates was different than that for the phytoplankton-only aggregates and therefore had different impact on the size spectra. As already discussed above, fragmentation in mineral-ballasted aggregates is triggered when the POC:mineral ratio drops below a certain threshold because of a diminution of particle stickiness. Particles removed from their aggregate as fragmentation progresses would then most probably be among the last that assembled— and thus not very small particles — since their attachment would be the least consolidated. This process may explain why the fragmentation in this case led to the formation of particles with intermediate sizes. Alternatively, the fragmentation or break-up of phytoplankton aggregates could be due to a POC loss that increased further its already high porosity and its fragility in any part of its structure. Small particles detaching from the larger one during fragmentation could then be of any size and potentially very small. These hypotheses are not easily tested in the frame of our experiment but Engel et al. (2009) who observed the formation of smaller aggregates with a calcifying strain of *Emiliana huxleyi* also reported contrasted fragmentation mechanisms between ballasted and non-ballasted aggregates. They attributed these differences to the role of protection that minerals could play against microbial

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744 remineralization — a mechanism still debated (Armstrong et al. 2002; Iversen and Ploug 2010; Iversen and Robert 2015) — and showed that aggregates with minerals could be better protected against fragmentation because of a more preserved organic cement. If true, this mechanism could suggest that ballast minerals would promote the formation of medium-sized particles with high density thereby serving as efficient carriers of carbon to the deep ocean.

749
Concluding remarks: implications for biogeochemical model parameterizations

The initial goal of this study was to ease SV parameterization in biogeochemical models by bringing evidence that Stokes' Law could reliably be applied to a certain category of marine aggregates (i.e. ballasted aggregates). Our results suggest that the SV of aggregated POM containing minerals could be parameterized in numerical models as a simple function of size and composition (i.e. relative fractions of organic and inorganic components, and porosity) using a Stokes' model. Although attempting such parameterization was beyond the scope of this paper we provide the following recommendations: (i) A Stokes' model parameterized with a fractal porosity (Eq. 7) should be used preferentially (model 3) to account for aggregate density variations with size. A fractal number $D_3$ of ~1.8 appears to be the most appropriate value since it adequately reproduces the slope of the size-SV relationship for the numerous datasets of ballasted aggregates considered here. It is important to note that a large range of fractal dimensions have been previously observed for marine aggregates (Kilps 1993; Maggi 2007) and this recommendation of a constant $D_3$ value is only intended here to facilitate sinking velocity parameterizations in biogeochemical models. At constant $D_3$ and for a given particle size range, porosity influencing aggregate density (and thus SV) is a function of the empirical constant $a$. Variations of $a$ between 0.1 and 0.001 reproduced the observed range of SV for aggregates carrying between ~7 and 98% of minerals (mineral:DW ratio). This observation suggests that $a$ could be empirically scaled to aggregate composition (especially to its mineral content), but additional field and experimental data are required to find an adequate formulation. If successful, this result may importantly support the 'ballast hypothesis' (Armstrong et al. 2002; Armstrong et al. 2009; Klaas and Archer 2002) currently debated over the absence of a clear relationship between particle SV and mineral load (Aumont et al. 2017; Lee et al. 2009).

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(ii) The applicability of our findings is likely to improve with increasing depth in the ocean due to the increase of the mineral:POM ratio through remineralization. This result implies that a parameterization of particle SV with a Stokes' model could be more robust in the mesopelagic layer and deeper ocean. Increase of particle SV with depth (Berelson 2002; Trull et al. 2008; Villa-Alfageme et al. 2016) could thus be potentially simulated through compositional changes in addition to depth-varying size classes (SV strictly size-dependent; Kriest and Oschlies 2008) or a prescribed linear SV increase with depth (Aumont et al. 2015; Gehlen et al. 2006; Kriest and Oschlies 2013).
781 Acknowledgements

782 We thank H. Stibor, Oceanopolis Brest, B. Becker, E. Grosstefan, M. Le Goff, P. Pondaven, A.

783 C. Baudoux, R. Corvaisier, A. Masson, C. Liourzou (IUEM), C. Labry (IFREMER) for technical support, discussions, and/or advice. We would also like to thank F. Maggi and one anonymous reviewer for their constructive comments and helpful insights. This work was supported by the CalMarO FP7 Marie Curie Initial Training Network (www.calmaro.eu, grant agreement no.

786 215157), Region Bretagne, EURO-BASIN (No. 264933), an Integrating Project of the European Community 7th Framework Program, and by the Australian Research Council through a Laureate awarded to Philip W. Boyd (FL160100131).

790
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