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.