Phosphorus model details

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A schematic of the modelled phosphorus cycle.

This article provides information on phosphorus model details, including the ordinary differential equations and parameters that describe the model.


The schematic to the right shows how the oceanic cycle of phosphorus is represented in the model. The model is a standard one-dimensional, two-box model of the global ocean, with the top layer representing the surface ocean down to the limit of the deepest wind-induced mixing during the year (down to the annual thermocline) and the bottom layer representing the deep ocean.

There is a simple representation of the physical processes that transfer phosphorus vertically in the ocean. This occurs in reality partly through mixing between the surface and deep oceans, partly through the physical movement of large masses of water between the surface and deep layers (upwelling and downwelling, ocean circulation), and partly due to diffusion. These physical processes are simulated by exchanging a slab of water between the surface and deep boxes. The rate of exchange equates to swapping a slab of thickness 3 metres every year (according to the standard value of the parameter K).

State variables and equations

The model is stripped down to the simplest form that can still reproduce the essential aspects of how phosphorus (in the form of phosphate) cycles in the oceans. There are only three state variables in my model: (1) the concentration of phosphate in the upper (surface) box (PS), (2) the concentration of phosphate in the lower (deep) box (PD), and (3) the concentration of phytoplankton in the surface box only (A). The ordinary differential equations for the three state variables are:


Equation 1:
The first term calculates uptake of phosphorus (as phosphate) by phytoplankton; the second represents remineralisation of organic material back to phosphate in the surface box; the third term is the net exchange of phosphate due to mixing between the surface and deep ocean boxes; and the fourth term represents riverine input of phosphate.

Equation 2:
The first term represents remineralisation of organic material back to phosphate in the deep box; and the second term is the net exchange of phosphate due to mixing between the deep and surface ocean boxes.

Equation 3:
The first term is new growth of algal biomass, determined according to phosphate limitation (PH is the Michaelis-Menten half-saturation constant that defines the way in which scarcity of phosphate impacts on growth rate); and the second term is the loss of algal cells due to mortality (M) processes such as grazing by zooplankton.

Model parameters

Other symbols in the equations above signify (1) the maximum possible growth rate of the phytoplankton (i.e. when phosphate is extremely plentiful) (μmax), (2) the fraction of dead phytoplankton that get decomposed (returned to inorganic nutrients) in the surface box (SR) and the deep box (DR) with the remainder being buried, (3) the amount of phosphorus coming down rivers (RP), and (4) the depths of the surface (SD) and deep (DD) boxes.

Phosphorus model parameters
Parameter Description Default
μmax maximum phytoplankton growth rate 0.25 d-1
M phytoplankton mortality rate 0.20 d-1
PH phosphate uptake half-saturation constant 0.03 mmol P m-3
SR surface box remineralisation fraction 0.95 -
DR deep box remineralisation fraction 0.048 -
SF sedimentation fraction 0.02 -
RP riverine supply of phosphate 0.2 mmol P m-2 y-1
SD surface box thickness 500 m
DD deep box thickness 3230 m
K surface-deep mixing rate 3 m y-1

The calculation of phytoplankton growth rates omits the effects of light, temperature and other nutrients on growth rate, all of which are important over some parts of the surface ocean at some times. To compensate, the phytoplankton are given lower maximum growth rates than laboratory and field experiments indicate should otherwise be the case.

Parameter values for the phosphorus model are the same as those used in the nitrogen-phosphorus model.

Further information

Further Reading: