Appendix L Processes: phyto losses

Once the respiration \(R_{resp\langle computed\rangle}^{phy}\) and exudation \(R_{exud\langle computed\rangle}^{phy}\) rates have been computed for a phytoplankton group via Equation (I.6) and Equation (I.12) respectively (and potentially Equation (I.7) if salinity limitation is activated), then those rates drive the mass losses of carbon, nitrogen, phosphorus and silicate. These losses are computed using a number of user specified (or default) parameters, and for each relevant computed variable, comprise the following:

• Excretion loss: Computed from a combination of respiration and exudation rates (\(R_{resp\langle computed\rangle}^{phy}\) and \(R_{exud\langle computed\rangle}^{phy}\), respectively), and
• Mortality loss: Computed from the respiration rate \(R_{resp\langle computed\rangle}^{phy}\)

The specifics of the calculation of each relevant computed variable’s excretion and mortality are described following. Losses are denoted as fluxes, \(F\).

L.1 Carbon

Equation (L.1) describes the calculation of excretion losses.

\[\begin{equation} \href{AppDiags.html#WQDiagPhyExcr}{F_{C-excr}^{phy}} = \left(\left[\left(1 - f_{true-resp}^{phy}\right) \times f_{excr-loss}^{phy} \times R_{resp\langle computed\rangle}^{phy}\right] + R_{exud\langle computed\rangle}^{phy}\right) \times \left[PHY\right] \tag{L.1} \end{equation}\]

\(F_{C-excr}^{phy}\) is the excretive loss of carbon and \(f_{true-resp}^{phy}\) is the fraction of respiration that corresponds to the generation of energy via consumption of stored chlorophyll a (i.e. the fraction that does not result in excretive or mortality losses). In Equation (L.1), (1 - \(f_{true-resp}^{phy}\)) is therefore the respiration that is associated with the combined excretive and mortality losses, rather than energy production. \(f_{excr-loss}^{phy}\) is the fraction of this combined loss that is excretive only, \(R_{resp\langle computed\rangle}^{phy}\) and \(R_{exud\langle computed\rangle}^{phy}\) are the computed respiration and exudation rates, respectively and \(\left[PHY\right]\) is a computational cell’s phytoplankton concentration.

Equation (L.2) describes the calculation of mortality losses of carbon.

\[\begin{equation} \href{AppDiags.html#WQDiagPhyMort}{F_{C-mort}^{phy}} = \left(\left(1 - f_{true-resp}^{phy}\right) \times\left(1 - f_{excr-loss}^{phy}\right) \times R_{resp\langle computed\rangle}^{phy}\right) \times \left[PHY\right] \tag{L.2} \end{equation}\]

\(F_{C-mort}^{phy}\) is the mortality loss of carbon and the other terms in Equation (L.2) as the same as Equation (L.1), noting the inclusion of (1 - \(f_{excr-loss}^{phy}\)) in Equation (L.2) to isolate the fraction of the combined excretive and mortality loss that is mortality only.

As with primary productivity, carbon does not need to be explicitly simulated as a computed variable for these losses to occur, but because carbon is used as the units of accounting for phytoplankton biomass, its losses are computed by the WQ Module in order to report phytoplankton concentrations. Carbon stores to which the phytoplankton losses in Equations (L.1) and (L.2) deliver are assumed to be unlimited when carbon is not explicitly included in a simulation as a computed variable.

The form of the entire multiplicative factors immediately preceding \(\left[PHY\right]\) in Equations (L.1) and (L.2) are presented in Figure L.1, as a function of \(f_{excr-loss}^{phy}\). Use the play button or drag the slider to see how different values of \(f_{true-resp}^{phy}\) change these multiplicative factors (ordinate), which are shown as functions of \(f_{excr-loss}^{phy}\) (abscissa). \(R_{resp\langle computed\rangle}^{phy}\) and \(R_{exud\langle computed\rangle}^{phy}\) are set to fixed values of 0.1 and 0.05 /day, respectively.

Figure L.1: Move the slider to see the effect of changing \(f_{true-resp}^{phy}\) on the form of the multiplicative factors preceding \([PHY]\) in the above loss equations. \(R_{resp<computed>}^{phy}\) and \(R_{exud< computed>}^{phy}\) are set to fixed values of 0.1 and 0.05 /day, respectively

To complement Figure L.1, the form of the entire multiplicative factors immediately preceding \(\left[PHY\right]\) in Equations (L.1) and (L.2) are again presented in Figure L.2, but as a function of \(f_{true-resp}^{phy}\). Use the play button or drag the slider to see how different values of \(f_{excr-loss}^{phy}\) change these multiplicative factors (ordinate), which are shown as functions of \(f_{true-resp}^{phy}\) (abscissa). \(R_{resp\langle computed\rangle}^{phy}\) and \(R_{exud\langle computed\rangle}^{phy}\) are again set to the same fixed values of 0.1 and 0.05 /day, respectively.

Figure L.2: Move the slider to see the effect of changing \(f_{excr-loss}^{phy}\) on the form of the multiplicative factors preceding \([PHY]\) in the above loss equations. \(R_{resp<computed>}^{phy}\) and \(R_{exud< computed>}^{phy}\) are set to fixed values of 0.1 and 0.05 /day, respectively

Some key features of the Figures L.1 and L.2 are as follows:

• If \(f_{true-resp}^{phy}\) is set to 1.0, then carbonaceous excretion losses are due solely to exudation, and mortality losses are zero. This is consistent with interpreting \(f_{true-resp}^{phy}\) as the fraction of respiration that is associated only with the re-release of energy through metabolism of carbon biomass generated and stored during antecedent phytoplanktonic photosynthesis
• Conversely, if \(f_{true-resp}^{phy}\) is set to 0.0, then (potentially unrealistically) respiration does not re-release any energy, but rather, sees the entire respiration rate expended on excretion of carbon biomass (subject to the value of \(f_{excr-loss}^{phy}\))
• If \(f_{excr-loss}^{phy}\) is set to 1.0, then mortality losses are zero. This is consistent with interpreting this parameter as the proportion of combined excretive and mortality losses that are excretive. The converse applies for a \(f_{excr-loss}^{phy}\) of 0.0

Intermediate behaviours to the above limits can be inferred through interacting with Figures L.1 and L.2.

L.2 Nitrogen

Equation (L.3) describes the calculation of excretion losses.

\[\begin{equation} \href{AppDiags.html#WQDiagPhyExcrN}{F_{N-excr}^{phy}} = \left(\left(f_{excr-loss}^{phy} \times R_{resp\langle computed\rangle}^{phy}\right) + R_{exud\langle computed\rangle}^{phy}\right) \times X_{N-C}^{phy} \times \left[PHY\right] \tag{L.3} \end{equation}\]

\(F_{N-excr}^{phy}\) is the excretive loss of nitrogen, \(f_{excr-loss}^{phy}\) is the fraction of combined excretive and mortality losses that is excretive only, \(R_{resp\langle computed\rangle}^{phy}\) and \(R_{exud\langle computed\rangle}^{phy}\) are the computed respiration and exudation rates, respectively, and \(\left[PHY\right]\) is a computational cell’s phytoplankton concentration. If the basic phytoplankton constituent model is used, then \(X_{N-C}^{phy}\) is the specified (or default) constant ratio of internal nitrogen to carbon. If the advanced phytoplankton constituent model is used, then \(X_{N-C}^{phy}\) is the dynamically computed ratio of internal nitrogen to carbon concentrations, where this ratio varies only between the specified (or default) minimum and maximum ratios \(X_{N-C-min}^{phy}\) and \(X_{N-C-max}^{phy}\), respectively.

Equation (L.4) describes the calculation of mortality losses.

\[\begin{equation} \href{AppDiags.html#WQDiagPhyMortN}{F_{N-mort}^{phy}} = \left(\left(1 - f_{excr-loss}^{phy}\right) \times R_{resp\langle computed\rangle}^{phy}\right) \times X_{N-C}^{phy} \times \left[PHY\right] \tag{L.4} \end{equation}\]

\(F_{N-mort}^{phy}\) is the mortality loss of nitrogen and the other terms in Equation (L.4) as the same as Equation (L.3), noting the inclusion of (1 - \(f_{excr-loss}^{phy}\)) in Equation (L.4) to isolate the fraction of the combined excretive and mortality loss that is mortality only.

The key feature of Equations (L.3) and (L.4) is that if \(f_{excr-loss}^{phy}\) is set to 1.0, then mortality losses are zero. This is consistent with interpreting this parameter as the proportion of combined excretive and mortality losses that are excretive. The converse applies for a \(f_{excr-loss}^{phy}\) of 0.0.

L.3 Phosphorus

Equation (L.5) describes the calculation of excretion losses.

\[\begin{equation} \href{AppDiags.html#WQDiagPhyExcrP}{F_{P-excr}^{phy}} = \left(\left(f_{excr-loss}^{phy} \times R_{resp\langle computed\rangle}^{phy}\right) + R_{exud\langle computed\rangle}^{phy}\right) \times X_{P-C}^{phy} \times \left[PHY\right] \tag{L.5} \end{equation}\]

\(F_{P-excr}^{phy}\) is the excretive loss of phosphorus, \(f_{excr-loss}^{phy}\) is the fraction of combined excretive and mortality losses that is excretive only, \(R_{resp\langle computed\rangle}^{phy}\) and and \(R_{exud\langle computed\rangle}^{phy}\) are the computed respiration and exudation rates, respectively, and \(\left[PHY\right]\) is a computational cell’s phytoplankton concentration. If the basic phytoplankton constituent model is used, then \(X_{P-C}^{phy}\) is the specified (or default) constant ratio of internal phosphorus to carbon. If the advanced phytoplankton constituent model is used, then \(X_{P-C}^{phy}\) is the dynamically computed ratio of internal phosphorus to carbon concentrations, where this ratio varies only between the specified (or default) minimum and maximum ratios \(X_{P-C-min}^{phy}\) and \(X_{P-C-max}^{phy}\), respectively.

Equation (L.6) describes the calculation of mortality losses.

\[\begin{equation} \href{AppDiags.html#WQDiagPhyMortP}{F_{P-mort}^{phy}} = \left(\left(1.0 - f_{excr-loss}^{phy}\right) \times R_{resp\langle computed\rangle}^{phy}\right) \times X_{P-C}^{phy} \times \left[PHY\right] \tag{L.6} \end{equation}\]

\(F_{P-mort}^{phy}\) is the mortality loss of phosphorus and the other terms in Equation (L.6) as the same as Equation (L.5), noting the inclusion of (1 - \(f_{excr-loss}^{phy}\)) in Equation (L.6) to isolate the fraction of the combined excretive and mortality loss that is mortality only.

As for nitrogen losses, the key feature of Equations (L.5) and (L.6) is that if \(f_{excr-loss}^{phy}\) is set to 1.0, then mortality losses are zero. This is consistent with interpreting this parameter as the proportion of combined excretive and mortality losses that are excretive. The converse applies for a \(f_{excr-loss}^{phy}\) of 0.0.

L.4 Silicate

Equation (L.7) describes the calculation of excretion losses, if silicate is included in phytoplanktonic calculations.

\[\begin{equation} F_{Si-excr}^{phy} = \left(\left(f_{excr-loss}^{phy} \times R_{resp\langle computed\rangle}^{phy}\right) + R_{exud\langle computed\rangle}^{phy}\right) \times X_{Si-C}^{phy} \times \left[PHY\right] \tag{L.7} \end{equation}\]

\(F_{Si-excr}^{phy}\) is the excretive loss of silicate, \(f_{excr-loss}^{phy}\) is the fraction of combined excretive and mortality losses that is excretive only, \(R_{resp\langle computed\rangle}^{phy}\) and and \(R_{exud\langle computed\rangle}^{phy}\) are the computed respiration and exudation rates, respectively, \(X_{Si-C}^{phy}\) is the specified (or default) constant ratio of internal silicate to carbon in the phytoplankton group being considered and \(\left[PHY\right]\) is a computational cell’s phytoplankton concentration.

Equation (L.8) describes the calculation of mortality losses.

\[\begin{equation} F_{Si-mort}^{phy} = \left(\left(1.0 - f_{excr-loss}^{phy}\right) \times R_{resp\langle computed\rangle}^{phy}\right) \times X_{Si-C}^{phy} \times \left[PHY\right] \tag{L.8} \end{equation}\]

Some key features of the Equations (L.7) and (L.8) are as follows:

• As for nitrogen and phosphorus losses, the key feature of Equations (L.7) and (L.8) is that if \(f_{excr-loss}^{phy}\) is set to 1.0, then mortality losses are zero. This is consistent with interpreting this parameter as the proportion of combined excretive and mortality losses that are excretive. The converse applies for a \(f_{excr-loss}^{phy}\) of 0.0
• Despite not simulating particulate silicate, Equations (L.7) and (L.8) nonetheless ensure that the mass of silicate delivered to the dissolved pool (that is simulated by the WQ Module) is consistent with that of carbon, nitrogen and phosphorus.