The constitutive elements of SLAMS are the aggregates. They are clusters of primary particles, a combination of phytoplankton cells (coccolithophorids, diatoms, picoplankton), TEP particles and terrigenous dust particles. SLAMS keeps track of a large number of aggregates that we call aggregate classes (ACs). Each AC carries a suite of information about the state of one class of aggregate in the water column (Table ). ACs, also called super-particles or super-droplets in atmospheric applications, are representative of some variable number, n, of identical aggregates (Sect. ). Each aggregate is composed of p primary particles (Fig. ), forming an aggregate consisting of up to 10 types of orgC (representing different ages), up to 10 types of TEP (also representing different ages) and four types of minerals (calcite, aragonite, biogenic silica and terrigenous material). Lability of orgC and TEP is determined from its age. From this information, SLAMS constructs the physical characteristics of the aggregate that determine its sinking velocity.

In all, 20 new ACs are created per 8 h time step and added to the list of existing ACs. This number is a trade-off between computation time and smoothness of the runs – 20 per time step turned out to be more than enough ACs to provide consistent results between runs and an acceptable run time. The type of phytoplankton (or terrigenous material) of each new particle is chosen by a Monte Carlo method, wherein a uniformly distributed random number x∼[0,1], is generated and compared with ranges in Table  (in this paper all random numbers are uniformly distributed on [0,1]). Table  shows the default functional group in SLAMS that is a typical open-ocean ensemble and would require modification with information on specific functional types and rates of particle production for detailed regional applications. When an AC is produced, p=1, meaning that the AC represents many copies of a single cell (primary particle) that has not aggregated. For example, if primary production is 1000 mg C m-2 d-1, each AC (unless it is terrigenous material) initially contains 1000/20⋅8/24=16.7 mg C or 1.39 mmol C. The amount of orgC per phytoplankton cell determines the number of primary particles (n) that the AC will initially represent (Table ). For example, a coccolithophorid in our model contains 7 pmol orgC. If the random number generated indicates production of coccolithophorids, n=1.39×10-3/7×10-12=1.98×108 coccolithophorids. The default functional groups (Table ) are used for all regions, except the Southern Ocean where calcifiers produce half as much calcium carbonate and diatoms produce twice as much biogenic silica as in the rest of the simulations.

The depth, z, of a new particle is determined by z=104⋅e-5⋅x, where x is a random number and z is in [cm]. Half the production thus takes place in the top 8.2 m.

Blooms are a condition of elevated phytoplankton concentration, possibly a result of complex predator–pray imbalance, and have confounded scientists for decades . Blooming diatoms have been found to have a lower transfer efficiency than a more carbonate dominated non-blooming flux, suggesting that compositional differences of aggregates and aggregate structure is important for understanding the controls on the organic carbon flux . Uncertainty in how spring blooms respond to changes in pCO2 and temperature are still unresolved . In SLAMS, the BI describes the degree to which production is characterized by blooms: BI=productivedays365, where BI = 1 represents constant production throughout the year and BI = 0.25 means that the annual production all takes place in a quarter of a year. It does not take into effect differences in ecology or physiology of the plankton that may be dominant in blooms vs. not in blooms. Continuing with the example above where PP = 1000 mg C-2 d-1, if BI = 0.5, then PP = 2000 mg C m-2 d-1 for the first half of the year and 0 for the second half of the year. Each AC produced in the first half of the year would contain 2000/20⋅8/24=33.4 mg C or 2.78 mmol C. Furthermore, if coccolithophorids are produced, n=2.78×10-3/7×10-12=3.96×108. No new ACs are produced during the second half of the year.

The kinetics of particle aggregation depend on the encounter rate and the probability of sticking: stickiness=α=interparticle attachment rateinterparticle collision rate . The main glue that holds marine snow together in the water column appears to be TEP, a mucus-like polysaccharide material exuded by phytoplankton and bacteria, especially under conditions of nutrient limitation during the senescent phase of phytoplankton blooms . It is exuded in dissolved form but it separates into suspended droplets, or gel, in conditions of turbulence . In SLAMS, TEP production consists of 6 % of the primary production in the default case, in terms of carbon. This number comes from sensitivity studies where we simulate equatorial Pacific conditions (SST, PP and seasonality) and compare orgC flux to the deep ocean to sediment trap data. In the model formulation, TEP is not part of primary production but an independent variable that can be changed without altering primary production, allowing for sensitivity analysis of TEP. In each time step, 20 new ACs are produced that are phytoplankton or terrigenous material, and 1 new AC that is TEP. In the example above, 6 % of primary production (1000 mg C m-2 d-1) is 60 mg C m-2 d-1. One TEP particle contains 1 pmol carbon and so n=5×109. The role of TEP in controlling the flux of organic matter is complicated however by its low density (∼ 0.8 g cm-3), which acts to decrease the overall density of an aggregate, potentially to the point where it becomes buoyant and ascends rather than sinks . In SLAMS, particles rich in TEP can be less dense than sea water resulting in upward movement. Occasional accumulation of TEP-rich aggregates at the sea surface prompted us to consider what happens to buoyant particles at the sea surface and include wind-driven surface mixing in the model.

As described above, much of the stickiness of aggregates appears to be due to TEP . Our formulation for particle stickiness is simple: TEP particles have α=1 and at the time of production the stickiness of all other particles is 0. After a particle aggregates, its stickiness is the volume-weighted average of the stickiness of the component particles: α=VTEPVa, where VTEP is the volume of TEP in the aggregate and Va is the volume of the entire aggregate. As a result, in SLAMS, particles do not stick unless TEP is present.

An aggregate that forms by aggregation of smaller particles forms a fractal structure with a dimension, DN, that describes its porosity: p∼raDN where p is the number of primary particles and ra is the radius of the aggregate . The closer DN is to 3, the more the structure fills up three-dimensional space, and the lower its porosity. The fractal dimension of marine snow has been inferred from measurements of aggregate properties such as settling velocity, porosity and size, to be in the range of 1.3 to 2.3 . The porosity of marine snow appears to be large, always above 0.9 and mostly closer to 0.99, increasing with the diameter of the aggregate . Fecal pellets are more compact, and thus their porosity is smaller, about 0.43–0.65 . The aggregate sinking velocity depends on the aggregate radius and porosity. We follow calculating radius and porosity from the fractal dimension and composition of the aggregate (Table ). The fractal dimension of aggregates in SLAMS is DN=2.0 . Fecal pellets are not fractal in nature, so we use the relationship between fecal pellet volume and sinking rate developed by and find that a porosity of 0.5 results in sinking velocity of model pellets that compare well to that relationship. The resulting model pellet density for 0.5 porosity is near the range of measured pellet density of 1.08–1.2, depending on the composition of the pellet . The aggregate radius, ra, is obtained from the fractal dimension and the radius of the primary particle, rp: ra=rp⋅p1/DN. The radius of the primary particle, rp, is calculated from its volume under the assumption that it is a perfect sphere: rp=3Vp4π3, where Vp, the volume of the primary particles in the aggregate, is calculated by dividing the total volume of material in the aggregate, Vm, by the number of primary particles: Vp=Vmp and Vm=∑iϵSmwiρiXi, where Xi is the concentration [mol] of substance i, mwi is the molar weight, ρi is the density and S=∑orgC,∑TEP,CaCO3,bSi,dust.

The porosity, ϕ, is related to the volume of individual particles that makes up the aggregate to the total volume of the aggregate: ϕ=1-p⋅VpVa=1-prpra3, where Va=43πra3 . The density of the aggregate, ρa, is calculated from the density of the material in the aggregate, ρs, the density of seawater, ρsw and the porosity, ϕ: ρa=(1-ϕ)ρs+ρsw.

Sinking velocities of particles range from a few up to hundreds of meters per day and have been found to increase with depth . The sinking orgC flux of particles in the ocean, binned according to sinking velocity of the particles, has been found to exhibit a bimodal distribution, with substantial fluxes from particles sinking at about 1 m d-1, and close to 1000 m d-1, but very little in between suggesting there are two types of sinking particles that make up the orgC sinking flux . In SLAMS, the aggregate sinking velocity: u=8ra(ρa-ρsw)g3ρswf(Re), is calculated using Stokes' law , where g is the gravity of Earth. For low Reynolds numbers where viscous forces are dominant, f(Re)=24/Re and Re=2rau/ν, where ν is kinematic viscosity. For large Reynolds numbers where turbulence starts to play a role, the drag coefficient is calculated using Whites' approximation: f(Re)=24/Re+6/(1+Re)+0.4 , which is valid for Re<2×104. Substituting f(Re) with White's approximation results in a nonlinear equation with u as the variable, which we solve for using Newtons' method. Model aggregates only sink vertically, there are no lateral currents. Aggregates reach high Re conditions when settling at a few hundred meters per day, depending on their size.

Within the range of salinity found in the ocean, the viscosity of seawater is mainly a function of temperature . We use a formula that fits empirical data of seawater viscosity at 35 ppt: ν=5×10-6exp⁡(2250/T) to find the viscosity, ν, at temperature T [K]. The change in viscosity with temperature has a large effect on the sinking velocity of particles. For the change in sea water temperatures from the sea surface to the sea floor in the tropics, the change in viscosity leads to a 50 % decrease in sinking velocity.

Bacteria are relevant to the biological pump for their role in degradation of orgC. Organic carbon is a chemically heterogeneous combination of proteins, lignin and cellulose, which vary in structure and degradability , and as it degrades, its heterogeneity increases. As the structure is altered, it becomes increasingly unrecognizable to enzymes, producing the observed decrease in reactivity . TEP degradation is treated the same way as orgC degradation. Respiration rates have been measured, both on natural aggregates collected in the ocean, and on aggregates formed in laboratory roller tanks from freeze-killed diatom ooze. Higher respiration rates and bacterial production were found in younger (1–3 days) rather than in older (7–14 days) aggregates, as the labile proteins were respired first, leaving the less labile material to be respired more slowly . Observed respiration rates of newly produced diatom aggregates range between 0.057 and 0.089 d-1 or 0.083 ± 0.034 d-1 , and generally decrease with time, from 0.09 initially to 0.05 d-1 3 days later .

To simulate the decrease in degradation rate with orgC age, we track 10 age bins for orgC in each aggregate, and construct a degradation rate that is both a function of age and temperature: dorgCl=k(l,T)⋅orgCldt, where l stands for the age bin and T is temperature [∘C]. The age of the orgC in bin l, agel, is 2l-2 to 2l-1 days and k(l,T)=0.1/agel⋅exp⁡(ln⁡2(T-30)/10). This relation assigns freshly produced orgC at the sea surface a degradation lifetime of a few days, and it prescribes a decrease in reaction rate with phytoplankton age following the observations of . The temperature dependence results in about a doubling of the reaction rate with 10 ∘C of warming, a moderate activation energy of about 45 kJ mol-1. Aging of orgC and TEP is accomplished by transferring orgCldt/(agel-agel-1) orgC (or TEP) from bin l into bin l+1 for l=1,…,9. Material does not age out of the top bin, l=10.

If the fraction of TEP in an aggregate is less than 0.02, it breaks into Pf fragments as described in Sect. .

The presence of slowly sinking small particles in the deep ocean attests to the effects of disaggregation or fragmentation of aggregates in the water column . In a sectional modeling study, treated disaggregation as driven by turbulent flow in the ocean. Laboratory experiments with diatom aggregates (a fragile form of marine snow) find that stresses in excess of those due to turbulent shear in the water column are required to break them, pointing to biological shear and grazing as the dominant breaking mechanism . The fluid stress required to break aggregates can be found near appendages of swimming zooplankton . The abundance of marine snow in surface waters appears to undergo daily cycles , as does concentration of zooplankton. observed that the mean size of aggregates decreased when the zooplankton Euphausia Pacifica were abundant. Recently, the idea that zooplankton may break aggregates to decrease their sinking velocity and increase their surface area to encourage bacteria to break down refractory organic carbon and enhance its nutritional value has been proposed as a mechanism to close the carbon budget in the twilight zone . While questions remain unanswered as to the exact effect zooplankton has on marine particles and flux, observations suggest that interaction between zooplankton and marine snow does take place. Zooplankton in our model have the ability to fragment and ingest aggregates and produce a range of fecal pellet sizes. The zooplankton encounter rate is the function in SLAMS that is least supported by science. It is a damped function of how much food there is available. The factor 10-4.5 is chosen by requiring the orgC export to be 100 mg C m-2 day-2 and the orgC flux to the seafloor to be about 2.5 mg C m-2 day-2 for the default forcing. Renc=10-4.5-1logFdt, where F=∑j=1ξ∑l=110orgCl,j is the amount of orgC (food) in each depth bin. For each AC, at each time step, a random number, r, is generated and compared to Renc. If r<Renc, the zooplankton do encounter the AC and either ingest it (all the aggregates it represents) or fragment it.

In addition to biological dissolution of CaCO3 in zooplankton guts, there appears to be significant dissolution of the more soluble CaCO3 mineral aragonite in the water column . A significant fraction of the CaCO3 production flux is composed of aragonite , produced for example by pteropods. Water column conditions reach undersaturation with respect to aragonite at a shallower water depth than for calcite. Biogenic silica (bSi), the other main biomineral in the ocean, is undersaturated throughout the ocean, and is distinguished by a strong temperature sensitivity of its dissolution rate, dissolving significantly faster in warm waters . In the deep ocean, carbonate dissolution is a function of the degree of undersaturation: dCaCO3(i)=ktCaCO3(i)⋅Ω⋅CaCO3(i)dt, where Ω=(1-[CO3=]in situ/[CO3=]sat)η is the saturation state of the sea water for calcite and aragonite. kt(1) = 5 day-1, kt(2) = 3.2 day-1, η = 4.5 for calcite and 4.2 for aragonite. The carbonate ion profile or concentration with depth, [CO3=]insitu, is prescribed and dissolution of calcium carbonate does not feed back on it.

For bSi dissolution, the dissolution rate is calculated as dbSi=ksize⋅Rd(T)⋅bSidt with temperature sensitivity from as Rd(T)=1.32×1016exp⁡(-11481/T) and a size-dependent term: ksize=bSibSi+1×10-10 to simulate the protective organic matrix or membrane that coats live diatoms and serves as protection to dissolution of frustules . We assume that non-aggregated diatom cells are alive and thus have a coating to protect them from dissolving. The sensitivity of the flux to ksize is investigated in Sect. .

The temperature is constant in the mixed layer. Below the mixed layer profiles are a linear interpolation from the SST to 4 ∘C at 1 km to 2 ∘C at 4 km. In the model, the temperature is constant over the course of the year.

The carbonate ion concentration is set to 220 µmol kg-1 at the sea surface and 80 µmol kg-1 below 1 km depth. It is linearly interpolated between the surface and 1 km .

The mixed layer is isothermal as described in Sect. . Particles in the mixed layer are assigned a random depth within the mixed layer each time step.

In a time step (Fig. ) a small number (default = 21) of new particles are produced near the sea surface and their attributes such as composition, sinking velocity and stickiness are set. The water column is divided into depth bins (dz = 10 m). For pairs of ACs within a particular depth range (bin), the probability of aggregation is calculated and compared to a random number to assess whether the pair should stick. If so, the attributes of both particles involved are updated, the more numerous class losing members to aggregation with the less numerous class, which aggregates entirely. The encounter rate of each AC to zooplankton is also calculated for each depth bin. Next, the model loops through every AC and checks if it encountered zooplankton, respires orgC and TEP, dissolves minerals, disintegrates if there is not enough TEP to hold it together, ages orgC and TEP, checks if the size is small and considered to be dissolved, calculates new sinking velocity and settles or ascends accordingly. When the AC reaches the seafloor, its content is cleared and memory is freed for a new aggregate.

The rational for using a Monte Carlo method over a deterministic method is the large number of chemical components that can be resolved without increasing the runtime of the coagulation function. The runtime of the sectional coagulation model with N chemical components can be as high as O(a⋅bN), where a and b are the number of size bins and composition-ratio bins; however, with simplifications it is possible to decrease the computational cost . In contrast, for a stochastic model, the runtime of an aggregation simulation is proportional to the number of ACs squared, and does not increase with N. This can be considerable and probably does not make SLAMS suitable for large ocean circulation models without simplifications.

Variations of the Monte Carlo method for aggregation developed by are used to study aggregation in astrophysics , atmospheric chemistry and oceanography . The approaches vary to fit the goal of each study, for example, in whether mass or number of ACs is conserved, and in how many true particles an AC represents.

One computational difference between the our model and some of these is that we use a super-droplet method , where each simulated particle (AC) represents a large and variable number of real particles. The super-droplet method bypasses computational problems that have historically been considered a great drawback of Monte Carlo methods. Another distinction is that our model allows for multi-stage aggregation within a given time step. The main focus of our model of sinking aggregates is tracking particles, their composition and geometry, as they aggregate and are chemically and biologically altered by bacteria, zooplankton and water chemistry.

To assess the validity of the aggregation method derived in Sect. , we first look at the evolution of the particle spectrum with time where particles are not allowed to sink and the collision kernel is set to constant. We compare this spectrum to the analytical solution as presented by (Fig. a). The Monte Carlo algorithm is able to reproduce the time evolution of the particle-size spectrum when a minimum of 100 aggregate classes are tracked. (For the full model runs described below, the number of computational particles reached about 30 000.) As the statistical significance of the larger particle-size class in the Monte Carlo model declines, its abundance is subject to random fluctuations, as seen in the large variations in the Monte Carlo model at t = 64 (Fig. a).

In the second test we allow the particles to sink and the curvilinear collision kernel used in SLAMS is used to determine aggregation. We look at the decrease of the mass remaining in the top box with time as also studied by (Fig. b). The similarity of the simulation results boosts our confidence in the Lagrangian model.

Particle-size distribution is a property of marine system; it provides information about the structure of the ecosystem and particle dynamics. The particle-size spectrum in the ocean has been measured using particle counters and imaging methods. The slope of the particle-size spectrum is usually calculated by dividing the concentration ΔC of particles in a given size range Δr by the size range: n=ΔC/Δr. A global synthesis of the particle-size spectrum found its slope to be in the range -2 to -5 and generally that low PP regions were associated with a steep spectrum, whereas greater productivity regions often exhibits a slope between -2 and -4. In a study confined to the Arabian Sea, the slope varied between -2 and -4 and again the greater negative values were generally found where PP was lower . In our model the slope of the whole spectrum varies around -3 (Fig. ).

To see how well the model captures the flux of material in the water column, we compare three model results to data: the orgC flux in the water column, the pe-ratio and the rain ratio at the sea floor. Sediment trap data exist in low resolution, temporally and spatially. Existing data have been analyzed quite extensively . Our comparison of data with the model is based on the synthesis of , who looked for correlations between primary production and export flux and the attenuation coefficient b in the Martin curve (Eq. ) of trap fluxes from 11 different regions of the world's oceans. Table  shows the values of model driving parameters for the 11 regions and Fig.  compares model results with data. Except for the Panama Basin, the model predicts the flux to the seafloor within a factor of 2–5 (Fig. ). The topography of the Panama Basin is complex and lateral transport of resuspended material has been observed, possibly explaining the high orgC flux seen in sediment traps .

In SLAMS, export is defined as flux out of the top 100 m. To see if the model captures relationships to do with export seen in the real world, model responses to sea surface temperature and rate of primary production are compared with data in Fig. . These data were compiled from field observations of primary production and particle export in . SLAMS produces the relationship between SST and export production quite well (Fig. a). The positive relationship seen in the date between export production and high PP is also produced by the model, but because in most of our test cases PP is relatively low this relationship is not demonstrated very clearly in Fig. d. No relationship is seen in the data between export production and SST, as well as between pe-ratio and PP. The model also sees no relationships between these pairs (Fig. b and c respectively).

Lastly, we compare the rain ratio (organic : inorgC) in the sinking carbon flux at the seafloor with rain ratio data . SLAMS reproduces the observed value of approximately 1 in the tropics with higher values in the subtropics and polar latitudes (Fig. ). The organic : inorgC ratio of primary production is the same in all model runs except for the Southern Ocean, run where biS production is doubled but CaCO3 production is halved.

The code is available at github: https://github.com/tinnsi/SLAMS.