Sinking flux of particulate organic matter in the oceans: Sensitivity to particle characteristics

The sinking of organic particles produced in the upper sunlit layers of the ocean forms an important limb of the oceanic biological pump, which impacts the sequestration of carbon and resupply of nutrients in the mesopelagic ocean. Particles raining out from the upper ocean undergo remineralization by bacteria colonized on their surface and interior, leading to an attenuation in the sinking flux of organic matter with depth. Here, we formulate a mechanistic model for the depth-dependent, sinking, particulate mass flux constituted by a range of sinking, remineralizing particles. Like previous studies, we find that the model does not achieve the characteristic ‘Martin curve’ flux profile with a single type of particle, but instead requires a distribution of particle sizes and/or properties. We consider various functional forms of remineralization appropriate for solid/compact particles, and aggregates with an anoxic or oxic interior. We explore the sensitivity of the shape of the flux vs. depth profile to the choice of remineralization function, relative particle density, particle size distribution, and water column density stratification, and find that neither a power-law nor exponential function provides a definitively superior fit to the modeled profiles. The profiles are also sensitive to the time history of the particle source. Varying surface particle size distribution (via the slope of the particle number spectrum) over 3 days to represent a transient phytoplankton bloom results in transient subsurface maxima or pulses in the sinking mass flux. This work contributes to a growing body of mechanistic export flux models that offer scope to incorporate underlying dynamical and biological processes into global carbon cycle models.


Modeling aggregates
In this theory, we use the fractal dimension, D, to represent the geometry of particles that are aggregates of material. Particle aggregates are most likely to form within the euphotic zone, where particle abundances are high, and the probability of particle collisions is enhanced through elevated shear and turbulence within the mixed layer 8 , by coagulation, or by the repackaging of organic material through trophic interactions, ingestion and egestion. 9 Aggregates, whose effective radius we denote by a g , are typically large particles composed of many loosely or tightly packed sub-particles (Fig. S2). They have a size-dependent volume to mass relationship that can be parameterized according to the particle fractal dimension, D 10 , that typically varies between 1 and 3. Fluffier, more porous aggregates have low values of D, while compact aggregates have a value of D that approaches 3 (the limit for a solid sphere).
Aggregates formed by Brownian motion, shear coagulation and differential sedimentation have fractal geometry 9 . This property allows a scaling between the fraction of sub-particles and the particle size f = A f ( a g a s ) D−3 , introducing two new variables; the fractal dimension D (must be < 3) and a constant A f 10 . Over the range of reported values of D for different aggregation experiments 11 we find that our results are not sensitive to this choice. Thus, we can write the Stokes sinking velocity for an aggregate and proceed in a similar manner as the solution given above.
An aggregate of radius a g is composed of many sub-particles, here assumed to be all of radius a s and density ρ p . We Figure S1. The a) number density and c) mass density of sinking particles predicted from (10) and (11)  now have a permeable particle sinking at velocity w, on which the drag force is reduced by a factor Ω given by Debye and Brinkman 12 as where β −2 is the non-dimensional permeability. The drag force on an aggregate is thus given by F d = 6 Ω π µ a g w.
Here we assume that although an aggregate has a large porosity 13 , the permeability (flow of water through the aggregate) is low, so β 1 and Ω 1. An aggregate of radius a g is composed of many sub-particles, here assumed to be all of radius a s and density ρ p .
If the aggregate were compressed into a solid ball, it would have a radius a and a gravity force F g . But, F d can also be written with an effective density ρ e f f as though the particle has diameter a g and its density is reduced because a fraction (1 − f ) of the particle volume is filled with the ambient fluid (we also neglect that the aggregate could entrain less dense water downward along with it, as it sinks 14 ).
The gravitational force on the particle is therefore given by where ρ e f f is Jackson 10 relates a to a g according to 2/4 a a a sub g Figure S2. Conceptual schematic of an aggregate with radius a g composed of sub-particles of radius a s and density ρ p . If the aggregate were to be compacted into a solid ball, the new radius would be a and the fraction of the new volume relative to the initial volume, f . The 'effective density' of the aggregate is therefore where A f and D are empirical constants (Jackson 10 uses A f = 1.49, D = 2.33). D is the 'fractal dimension' (for a solid object, D=3). So, the volume fraction ( f ) becomes Substituting f into (3) and (4) we get Equating the two forces and solving for the terminal velocity of the aggregate Equation (8) is integrable by separation of parts and is combined with remineralization a g = a 0 (1 − rt) to arrive at the solution for the aggregate radius a g as a function of depth, where The void space within the particles must be accounted when computing the mass flux F M agg for aggregates with a range of sizes. This can be accomplished be multiplying the volume V n g = 4πa 3 g 3 of each size class n by its 'effective' density ρ e f f . Thus, for N n particles of size class n, the mass flux (per m 2 per day) is given by ρ n e f f N n V n g (z), for z eu ≤ z d ≤ z p dis 0, for z d > z p dis .
The total mass flux is again the sum over all size classes and is given by 3/4