## Examples Of Simple Models Relevant To Planktonic Systems

13.3.1 Rectilinear, Monodisperse, and Volume Conserving

13.3.1.1 Phytoplankton and the Critical Concentration

The original model of Jackson22 considered an algal population in the surface mixed layer as consisting of single cells whose number concentration Cl increased as the cells divided with a specific growth rate \x and which disappeared as they fell out of a mixed layer and as they collided to form aggregates:

where a is the stickiness, Z is the mixed layer thickness, and v1 is the settling velocity of a particle composed of one algal cell. Concentrations of aggregates containing j algal cells increased and decreased with aggregation and sinking:

for j > 1. Note that the index (j — i) is used to indicate that a particle with j monomers requires that the second particle in a collision have (j — i) monomers if the first particle has i monomers. The original model used rectilinear kernels, initially monodisperse particle sources and mass-length relationships akin to fractal scaling.

Simulation results of such a system show that this is essentially a two-state system (Figure 13.1; parameter values in figure caption). For the first 3 days, the only particle size class to change is that of single algal cells, which increases exponentially (linear in a logarithmic axis); larger particles have essentially constant concentrations. With time, ever large particles have their concentrations changed. For the particles composed of 30 monomers, there is an increase in concentration of about 10 orders of magnitude between days 6 and 9. After day 9, there is essentially no change in concentration. The difference in the first 3 days and the period after day 6 can be understood as resulting from very few formation of aggregates at low algal concentrations, but formation of aggregates at a rate that matches algal division at higher monomer concentrations. The rapid aggregate formation blocks any further increase in algal numbers despite continued cell production.

The limitation can be understood by simplifying Equation (13.11) and assuming that the most important loss for single cells is to collision and subsequent coagulation

FIGURE 13.1 Number concentration of particles for an exponentially growing algal population as a function of number of algal cells in a particle and time. The single algal cell with a radius of r1 = 10 /m and stickiness of a = 1 grows exponentially at / = 1 per days in a Z = 30 m thick mixed layer having a shear of y = 1 sec1. Particle fall velocity is calculated using a particle density of 1.036 g cm-3 and fluid density of 1.0 g cm-3. The calculation uses the summation formulation of Equations (13.11 and 13.12) and a rectilinear coagulation kernel.

FIGURE 13.1 Number concentration of particles for an exponentially growing algal population as a function of number of algal cells in a particle and time. The single algal cell with a radius of r1 = 10 /m and stickiness of a = 1 grows exponentially at / = 1 per days in a Z = 30 m thick mixed layer having a shear of y = 1 sec1. Particle fall velocity is calculated using a particle density of 1.036 g cm-3 and fluid density of 1.0 g cm-3. The calculation uses the summation formulation of Equations (13.11 and 13.12) and a rectilinear coagulation kernel.

with other single cells:

dC1 2

di 1

where j11 = 1.3y (r1 + r1)3 (the rectilinear shear kernel). Note that the differential sedimentation kernel for collisions between two particles of the same size is zero because they fall at the same rate, and that the Brownian kernel is considerably smaller than that for shear for particles larger than 1 ^.m. At steady state, the generation of new algal cells by division balances the loss to coagulation. The resulting concentration for the cells is aßn = 1.3ay 8r3

Expressed as a volume concentration for spherical particles, this is:

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