# Appendix K Processes: phyto uptake

Once the primary productivity \(R_{prod\langle computed\rangle}^{phy}\) rate has been computed for a phytoplankton group via Equation (I.2) (and potentially Equations (I.3) and (I.4) if nitrogen fixing and salinity limitation are activated, respectively), then that rate drives the mass uptake of inorganic carbon, nitrogen, phosphorus and silicate. This uptake includes the operation of a number of user specified (or default) parameters, and the respective calculations are described following. Uptakes are denoted as fluxes, \(F\).

## K.1 Carbon

Carbon has the simplest uptake calculation, as per Equation (K.1).

\[\begin{equation} F_{C-uptake}^{phy} = R_{prod\langle computed\rangle}^{phy} \times \left[PHY\right] \tag{K.1} \end{equation}\]

\(F_{C-uptake}^{phy}\) is the uptake of carbon, \(R_{prod\langle computed\rangle}^{phy}\) is the computed primary productivity rate and \(\left[PHY\right]\) is a computational cell’s phytoplankton concentration. This flux is summed over all simulated phytoplankton groups to compute the community carbon uptake, \(F_{C-uptake\langle computed\rangle}^{comm}\).

Carbon does not need to be explicitly simulated as a computed variable for this uptake to occur, but because carbon is used as the units of accounting for phytoplankton biomass, its uptake is computed by the WQ Module in order to report phytoplankton concentrations. These concentrations are either mmol C per m\(^3\) or \(\mu\)g chlorophyll *a* per litre (using a specified or default conversion), depending on the specified simulation units system. Carbon stores from which phytoplankton uptake can draw are assumed to be unlimited and unaccounted for when carbon is not explicitly included in a simulation as a computed variable.

## K.2 Nitrogen

The WQ Module allows for phytoplankton to uptake nitrate and ammonium (i.e. inorganic nitrogen) to meet primary production nitrogen demands. The parallel uptake of organic nitrogen (i.e. simulation of mixotrophic phytoplankton) is not yet implemented within the WQM. Uptake is computed differently for the basic and advanced phytoplankton constituent models

### K.2.1 Basic phytoplankton constituent model

If internal phytoplankton nutrients are not simulated dynamically, nitrogen uptake is calculated as per Equation (K.2).

\[\begin{equation} F_{N-uptake}^{phy} = R_{prod\langle computed\rangle}^{phy} \times \left[PHY\right] \times X_{N-C-con}^{phy} \tag{K.2} \end{equation}\]

\(F_{N-uptake}^{phy}\) is the uptake of water column inorganic nitrogen, \(R_{prod\langle computed\rangle}^{phy}\) is the computed primary productivity rate, \(\left[PHY\right]\) is a computational cell’s phytoplankton concentration and \(X_{N-C-con}^{phy}\) is the specified (or default) constant ratio of internal nitrogen to carbon in the phytoplankton group being considered.

This quantity \(F_{N-uptake}^{phy}\) is the total uptake of inorganic nitrogen. In order to disaggregate this uptake into that of ammonium and nitrate, the WQ Module applies Equation (K.3), for ammonium concentrations greater than zero.

\[\begin{equation} \left.\begin{aligned} f_{amm-uptake}^{phy} =& \frac{\left[NH_4\right]\times\left[NO_3\right]}{\left(\left[NH_4\right] + K_{lim-N}^{phy}\right)\times\left(\left[NO_3\right]+K_{lim-N}^{phy}\right)} + \frac{\left[NH_4\right]\times K_{lim-N}^{phy}}{\left(\left[NH_4\right] + \left[NO_3\right]\right)\times\left(\left[NO_3\right]+K_{lim-N}^{phy}\right)} \\ \\ F_{amm-uptake}^{phy} =& F_{N-uptake}^{phy} \times f_{amm-uptake}^{phy} \\ \\ F_{nit-uptake}^{phy} =& F_{N-uptake}^{phy} \times \left(1.0 - f_{amm-uptake}^{phy}\right) \end{aligned}\right\} \tag{K.3} \end{equation}\]

For a phytoplankton group, \(f_{amm-uptake}^{phy}\) is the fraction of nitrogen uptake that is ammonium, \(\left[NH_4\right]\) and \(\left[NO_3\right]\) are the ambient water column ammonium and nitrate concentrations, respectively, \(K_{lim-N}^{phy}\) is the half saturation nitrogen concentration for phytoplankton uptake and \(F_{amm-uptake}^{phy}\) and \(F_{nit-uptake}^{phy}\) are the uptake by phytoplankton of ammonium and nitrate, respectively. These fluxes (whether computed from Equation (K.3) or (K.4)) are summed over all simulated phytoplankton groups to compute the community ammonium and nitrate uptake, \(F_{amm-uptake\langle computed\rangle}^{comm}\) and \(F_{nit-uptake\langle computed\rangle}^{comm}\), respectively.

The form of \(f_{amm-uptake}^{phy}\) in Equation (K.3) with ammonium concentration is presented in Figure K.1. Use the “play” button or drag the slider to see how different half saturation concentrations \(K_{lim-N}^{phy}\) change the fraction of ammonium taken up by phytoplankton (ordinate), as a function of ambient ammonium concentration (abscissa). Nitrate concentration is set to a fixed 5 mg/L.

To complement Figure K.1, the form of \(f_{amm-uptake}^{phy}\) in Equation (K.3) with nitrate concentration is presented in Figure K.2, using the same ordinate axis limits. Use the “play” button or drag the slider to see how different half saturation concentrations \(K_{lim-N}^{phy}\) change the fraction of ammonium taken up by phytoplankton (ordinate), as a function of ambient nitrate concentration (abscissa). Ammonium concentration is set to a fixed 2 mg/L.

The parameter \(K_{lim-N}^{phy}\) is the same as that used for calculation of the nitrogen limitation function in the basic (and advanced if internal nitrogen stores are exhausted) phytoplankton constituent model, so its specification should be considered in terms of its influence on both uptake on limitation.

### K.2.2 Advanced phytoplankton constituent model

If internal phytoplankton nutrients are simulated dynamically, nitrogen uptake is calculated as per Equation (K.4).

\[\begin{equation} F_{N-uptake}^{phy} = R_{N-uptake}^{phy} \times L_{T}^{phy} \times \left[PHY\right] \times \frac{\left(X_{N-C-max}^{phy} - \frac{\left[IN\right]}{\left[PHY\right]}\right)}{\left(X_{N-C-max}^{phy} - X_{N-C-min}^{phy}\right)} \times \frac{\left(\left[N\right]_{avail} - \left[N\right]_{min}\right)}{\left(\left[N\right]_{avail} - \left[N\right]_{min}\right) + K_{lim-N}^{phy}} \tag{K.4} \end{equation}\]

\(F_{N-uptake}^{phy}\) is the uptake of water column inorganic nitrogen, \(R_{N-uptake}^{phy}\) is the specified (or default) rate of uptake of inorganic nitrogen, \(L_{T}^{phy}\) is the temperature limitation function, \(\left[PHY\right]\) and \(\left[IN\right]\) are a computational cell’s phytoplankton and internal nitrogen concentrations respectively, \(X_{N-C-min}^{phy}\) and \(X_{N-C-max}^{phy}\) are the specified (or default) minimum and maximum ratios of internal nitrogen to carbon in the phytoplankton group being considered, respectively, \(\left[N\right]_{avail}\) is the available (water column) nitrogen pool on which phytoplankton can draw for primary productivity, and is the sum of inorganic (ammonium and nitrate) nitrogen, \(\left[N\right]_{min}^{phy}\) is the nitrogen concentration below which phytoplankton is no longer permitted to uptake nitrogen and \(K_{lim-N}^{phy}\) is the half saturation nitrogen concentration for phytoplankton uptake. These last three parameters (and the last term in Equation (K.4)) are the same as that applied to the basic phytoplankton constituent model as per Equation (K.2), via calculation of the limitation function applied to \(R_{prod\langle computed\rangle}^{phy}\) (as per Equation (J.9)).

Equation (K.4) reveals the following:

- The uptake of water column nitrogen to internal phytoplankton stores is governed by the specified (or default) nitrogen uptake rate
- This rate is modified by
- Temperature, using the specified (or default) temperature limitation model for uptake (Section J.2)
- The current internal nitrogen concentrations, relative to the specified (or default) minima and maxima
- Ambient water column inorganic nitrogen concentrations, and
- The half saturation nitrogen concentration for uptake, which is the same parameter as used in the basic phytoplankton constituent model

If for example the internal nitrogen concentration \(\left[IN\right]\) at a particular timestep and computational cell is equal to \(X_{N-C-max}^{phy} \times \left[PHY\right]\), then Equation (K.4) has that internal nitrogen stores are full and that uptake is therefore zero. Conversely, if \(\left[IN\right]\) at a particular timestep and computational cell is equal to \(X_{N-C-min}^{phy} \times \left[PHY\right]\), then internal nitrogen stores are at their lowest and therefore (in themselves) do not limit uptake of water column nitrogen. Finally, if the available water column nitrogen \(\left[N\right]_{avail}\) is equal to the specified (or default) minimum allowable nitrogen concentration for uptake \(\left[N\right]_{min}^{phy}\), then uptake to internal nutrient stores is zero.

If uptake from the water column to internal stores occurs, then the split between ammonium and nitrate uptake is computed as per Equation (K.3).

The parameter \(K_{lim-N}^{phy}\) is the same as that used for calculation of the nitrogen limitation function in the advanced phytoplankton constituent model if internal nitrogen stores are exhausted, so its specification should be considered in terms of its influence on both uptake on limitation.

### K.2.3 Nitrogen fixing

If nitrogen fixing by phytoplankton is simulated, then a modification to the above computed nitrogen uptakes (for both the basic and advanced phytoplankton constituent models) is applied. The flux of atmospheric nitrogen is computed as per Equation (K.5).

\[\begin{equation} F_{N-fix}^{phy} = R_{N-fix}^{phy} \times \left(1.0 - L_{nit}^{phy}\right) \times \left[PHY\right] \tag{K.5} \end{equation}\]

\(F_{N-fix}^{phy}\) is the fixing of atmospheric nitrogen, \(R_{N-fix}^{phy}\) is the specified (or default) rate of nitrogen fixing, \(L_{nit}^{phy}\) is the nitrogen limitation function computed via either Equation (J.9) or (J.11) for the basic and advanced phytoplankton constituent models, respectively, before \(L_{nit}^{phy}\) is set to one. Once computed, \(F_{N-uptake}^{phy}\) is modified as follows:

- If \(|F_{N-fix}^{phy}| \ge |F_{N-uptake}^{phy}|\) then all uptake is assigned to \(F_{N-fix}^{phy}\), and \(F_{N-uptake}^{phy}\) is set to zero.
- If If \(|F_{N-fix}^{phy}| \le |F_{N-uptake}^{phy}|\) then nitrogen water column uptake is reduced as per Equation (K.6).

\[\begin{equation} F_{N-uptake\langle computed \rangle}^{phy} = F_{N-uptake}^{phy} \times \frac{\left(|F_{N-uptake}^{phy}| - F_{N-fix}^{phy}\right)}{|F_{N-uptake}^{phy}|} \tag{K.6} \end{equation}\]

Equation (K.6) is a simple linearisation that maps \(F_{N-uptake\langle computed \rangle}^{phy}\) to a value between 0.0 (where \(F_{N-fix}^{phy} = F_{N-uptake}^{phy}\)) and \(F_{N-uptake\langle computed \rangle}^{phy}\) (where \(F_{N-fix}^{phy}\) is zero).

## K.3 Phosphorus

The WQ Module allows for phytoplankton to uptake filterable reactive phosphorus (i.e. inorganic phosphorus, FRP) to meet primary production phosphorus demands. The parallel uptake of organic phosphorus (i.e. simulation of mixotrophic phytoplankton) is not yet implemented within the WQM. Uptake is computed differently for the basic and advanced phytoplankton constituent models

### K.3.1 Basic phytoplankton constituent model

If internal phytoplankton nutrients are not simulated dynamically, phosphorus uptake is calculated as per Equation (K.7).

\[\begin{equation} F_{P-uptake}^{phy} = R_{prod\langle computed\rangle}^{phy} \times \left[PHY\right] \times X_{P-C-con}^{phy} \tag{K.7} \end{equation}\]

\(F_{P-uptake}^{phy}\) is the uptake of inorganic phosphorus, \(R_{prod\langle computed\rangle}^{phy}\) is the computed primary productivity rate, \(\left[PHY\right]\) is a computational cell’s phytoplankton concentration and \(X_{P-C-con}^{phy}\) is the specified (or default) constant ratio of internal phosphorus to carbon in the phytoplankton group being considered. This flux (whether computed from Equation (K.7) or (K.8)) is summed over all simulated phytoplankton groups to compute the community FRP uptake, \(F_{P-uptake\langle computed\rangle}^{comm}\).

### K.3.2 Advanced phytoplankton constituent model

If internal phytoplankton nutrients are simulated dynamically, phosphorus uptake is calculated as per Equation (K.8).

\[\begin{equation} F_{P-uptake}^{phy} = R_{P-uptake}^{phy} \times L_{T}^{phy} \times \left[PHY\right] \times \frac{\left(X_{P-C-max}^{phy} - \frac{\left[IP\right]}{\left[PHY\right]}\right)}{\left(X_{P-C-max}^{phy} - X_{P-C-min}^{phy}\right)} \times \frac{\left(\left[P\right]_{avail} - \left[P\right]_{min}^{phy}\right)}{\left(\left[P\right]_{avail} - \left[P\right]_{min}^{phy}\right) + K_{lim-P}^{phy}} \tag{K.8} \end{equation}\]

\(F_{P-uptake}^{phy}\) is the uptake of water column FRP, \(R_{P-uptake}^{phy}\) is the specified (or default) rate of uptake of FRP, \(L_{T}^{phy}\) is the temperature limitation function, \(\left[PHY\right]\) and \(\left[IP\right]\) are a computational cell’s phytoplankton and internal phosphorus concentrations respectively, \(X_{P-C-min}^{phy}\) and \(X_{P-C-max}^{phy}\) are the specified (or default) minimum and maximum ratios of internal phosphorus to carbon in the phytoplankton group being considered, respectively, \(\left[P\right]_{avail}\) is the available (i.e. ambient water column) FRP pool on which phytoplankton can draw for primary productivity, \(\left[P\right]_{min}^{phy}\) is the FRP concentration below which phytoplankton is no longer permitted to uptake phosphorus and \(K_{lim-P}^{phy}\) is the half saturation FRP concentration for phytoplankton uptake. These last three parameters (and the last term in Equation (K.8)) are the same as that applied to the basic phytoplankton constituent model as per Equation (K.7), via calculation of the limitation function applied to \(R_{prod\langle computed\rangle}^{phy}\) (as per Equation (J.13)).

Equation (K.8) reveals the following:

- The uptake of water column FRP to internal phytoplankton stores is governed by the specified (or default) phosphorus uptake rate
- This rate is modified by
- Temperature, using the specified (or default) temperature limitation model for uptake (Section J.2)
- The current internal phosphorus concentrations, relative to the specified (or default) minima and maxima
- Ambient water column FRP concentrations, and
- The half saturation phosphorus concentration for uptake, which is the same parameter as used in the basic phytoplankton constituent model

If for example the internal phosphorus concentration \(\left[IP\right]\) at a particular timestep and computational cell is equal to \(X_{P-C-max}^{phy}\times \left[PHY\right]\), then Equation (K.8) has that internal phosphorus stores are full and that uptake is therefore zero. Conversely, if \(\left[IP\right]\) at a particular timestep and computational cell is equal to \(X_{P-C-min}^{phy} \times \left[PHY\right]\), then internal phosphorus stores are at their lowest and therefore (in themselves) do not limit uptake of water column FRP. Finally, if the available water column FRP \(\left[P\right]_{avail}\) is equal to the specified (or default) minimum allowable FRP concentration for uptake \(\left[P\right]_{min}^{phy}\), then uptake to internal nutrient stores is zero.

The parameter \(K_{lim-P}^{phy}\) is the same as that used for calculation of the phosphorus limitation function in the advanced phytoplankton constituent model if internal phosphorus stores are exhausted, so its specification should be considered in terms of its influence on both uptake on limitation.

## K.4 Silicate

The WQ Module allows for phytoplankton to uptake silicate to meet primary production demands, for phytoplankton groups that require silicate. Silicate uptake is calculated as per Equation (K.9).

\[\begin{equation} F_{Si-uptake}^{phy} = R_{prod\langle computed\rangle}^{phy} \times \left[PHY\right] \times X_{Si-C-con}^{phy} \tag{K.9} \end{equation}\]

\(F_{Si-uptake}^{phy}\) is the uptake of silicate, \(R_{prod\langle computed\rangle}^{phy}\) is the computed primary productivity rate, \(\left[PHY\right]\) is a computational cell’s phytoplankton concentration and \(X_{Si-C-con}^{phy}\) is the specified (or default) constant ratio of internal silicate to carbon in the phytoplankton group being considered.