J.4 Phosphorus

Phosphorus offers two limitation functions, one for each of the basic and advanced phytoplankton constituent models. These are described separately following.

J.4.1 Basic phytoplankton constituent model

If internal phytoplankton nutrients are not dynamically simulated, Equation (J.13) is used to compute \(L_{phs}^{phy}\).

\[\begin{equation} L_{phs}^{phy} = \frac{\left(\left[P\right]_{avail} - \left[P\right]_{min}^{phy}\right)}{\left(\left[P\right]_{avail} - \left[P\right]_{min}^{phy}\right) + K_{lim-P}^{phy}} \tag{J.13} \end{equation}\]

\(\left[P\right]_{avail}\) is the available phosphorus pool on which phytoplankton can draw for primary productivity, and is simply FRP. \(\left[P\right]_{min}^{phy}\) is the phosphorus concentration below which phytoplankton no longer is permitted to uptake phosphorus \(K_{lim-P}^{phy}\) is the half saturation phosphorus concentration for phytoplankton uptake.

Equation (J.13) has that when \(K_{lim-P}^{phy}\) is equal to the difference between ambient phosphorus concentration and \(\left[P\right]_{min}^{phy}\), i.e. when

\[\begin{equation} K_{lim-P}^{phy} = \left[P\right]_{avail} - \left[P\right]_{min}^{phy} \tag{J.14} \end{equation}\]

then \(L_{phs}^{phy}\) = 0.5. The form of \(L_{phs}^{phy}\) with varying \(\left[P\right]_{min}^{phy}\) and \(K_{lim-P}^{phy}\) is provided in Figure J.4, for constant \(\left[P\right]_{avail}\) = 1.0 mg/L.

Figure J.4: Move the slider to see the effect of changing \(K_{lim-P}^{phy}\) on the computed phosphorus limitation function

J.4.2 Advanced phytoplankton constituent model

If internal phytoplankton nutrients are simulated dynamically, and phytoplankton concentrations are greater than the specified (or default) minimum, then Equation (J.15) is used to compute \(L_{phs}^{phy}\).

\[\begin{equation} L_{phs}^{phy} = \frac{X_{P-C-max}^{phy} \times \left(1.0 - \frac{X_{P-C-min}^{phy} \times \left[PHY\right] }{\left[IP\right]}\right)}{X_{P-C-max}^{phy} - X_{P-C-min}^{phy}} \tag{J.15} \end{equation}\]

\(X_{P-C-min}^{phy}\) and \(X_{P-C-max}^{phy}\) are the specified (or default) minimum and maximum internal phosphorus to Chl a (or carbon if using mmm units) ratios, respectively and \(\left[PHY\right]\) and \(\left[IP\right]\) are the current internal Chla a (or carbon) and phosphorus concentrations in the computational cell being considered, respectively. Substituting the definition for the minimum phosphorus to Chl a (or carbon) ratio \(X_{P-C-min}^{phy}=\left[IP\right]_{min}/\left[PHY\right]\) (where \(\left[PHY\right]\) is in either mgL or mmm units) into Equation (J.15) and rearranging, reveals the form of \(L_{phs}^{phy}\) as per Equation (J.16).

\[\begin{equation} L_{phs}^{phy} = \frac{X_{P-C-max}^{phy} \times \left(1.0 - \frac{\left[IP\right]_{min}} {\left[IP\right]}\right)}{X_{P-C-max}^{phy} - X_{P-C-min}^{phy}} \tag{J.16} \end{equation}\]

\(\left[IP\right]_{min}\) is the minimum internal phosphorus concentation corresponding to the current \(\left[PHY\right]\). Equation (J.16) demonstrates that:

• If an internal phosphorus concentration is equal to \(\left[IP\right]_{min}\), then \(L_{phs}^{phy}\) is zero. This means that no internal phosphorus is available for primary production in the current timestep and that phosphorus is therefore completely preventing growth
• If an internal phosphorus concentration is equal to \(\left[IP\right]_{max}\), then \(L_{phs}^{phy}\) is one. This means that maximum internal phosphorus is available for primary production in the current timestep and that phosphorus is therefore placing no limit on growth

If internal phytoplankton nutrients are simulated dynamically, and phytoplankton concentrations are less than the specified (or default) minimum, then phytoplankton will look to external (i.e. ambient water column) phosphorus pools to support primary productivity. The associated limitation function is therefore computed in the same manner as Equation (J.13) in Appendix J.4.1. If this is the case, then the parameters \(K_{lim-P}^{phy}\) and \(\left[P\right]_{min}^{phy}\) are used.

This same parameter is also used in the computation of the phosphorus that is taken up with primary production (Appendix K.3.2) in the advanced phytoplankton constituent model (Equation (K.8)). As such its setting should be considered in terms of its impacts on both phosphorus limitation and uptake processes in the advanced phytoplankton constituent model.