\(\newcommand{\p}[1]{\frac{\partial }{\partial #1}}\) \(\newcommand{\pp}[2]{\frac{\partial #1}{\partial #2}}\) \(\newcommand{\dd}[2]{\frac{d #1}{d #2}}\) \(\newcommand{\h}{\frac{1}{2}}\) \(\newcommand{\op}[1]{\operatorname{#1}}\)

8.7.3.3. Internal carbon store and exudation

Note: the internal carbon store is currently only implemented for fixed plankton elemental ratios (i.e., DARWIN_ALLOW_NQUOTA, DARWIN_ALLOW_PQUOTA, DARWIN_ALLOW_FEQUOTA, DARWIN_ALLOW_SIQUOTA all undefined).

With DARWIN_ALLOW_CSTORE defined, nutrient availability limits growth directly rather than via the photosynthesis rate. The maximum photosynthesis rate (and with it carbon uptake) becomes independent of nutritient limitation,

\[P^{{\mathrm{C}}{\op{m}}}_j = P_{{\mathrm{C}},j}^{\max} f^{{{\text{phy}}}}_j(T) \gamma_{\op{pCO2}}\]

but the growth rate is limited by the availability of each nutrient,

\[\mu_j = \min(P^{\op{C}}_j, \mu^{\max\op{N}}_j, \mu^{\max\op{P}}_j, \mu^{\max\op{Fe}}_j, \mu^{\max\op{Si}}_j)\]

Each nutrient’s availability determines a maximum achievable growth rate. Its form depends on whether variable quotas are enabled for the nutrient. For instance for phosphorus, without P quota,

(8.56)\[\mu^{\max\op{P}}_j = \frac{1}{R^{\op{P}:\op{C}}_j} V^{{\mathrm{P}}\max}_j \gamma^{\op{P}}_j f^{\op{up}}_j(T) ( 1 + r^{\op{resp}}_j )\]

and with P quota,

(8.57)\[\mu^{\max\op{P}}_j = \frac{1}{R^{\op{P}:\op{C}}_j} u^{\mathrm{P}}_j ( 1 + r^{\op{resp}}_j )\]

where \(u^{\mathrm{P}}_j\) is the carbon-specific uptake rate for phosphorus,

\[u^{\mathrm{P}}_j = V^{{\mathrm{P}}\max}_j \frac{\mathrm{PO}_4}{\mathrm{PO}_4 + k^{\op{PO4}}_j} {{\text{reg}}}^{Q{\mathrm{P}}}_j \cdot f^{{\text{up}}}_j(T)\]

and the respiration rate is

\[r^{\op{resp}}_j = r^{\op{resp}\max}_j f^{\op{remin}}(T) \frac{c_j - c_j^{\min}}{c_j}\]

Other elements work analogously, except that the maximum nitrogen uptake rate for the case without nitrogen quotas is more complex,

\[\mu^{\max\op{N}}_j = \frac{1}{R^{\op{P}:\op{C}}_j} u^{{\mathrm{N}}\max}_j ( 1 + r^{\op{resp}}_j )\]

where

\[u^{{\mathrm{N}}\max}_j = \max( u^{{\mathrm{NO3}}\max}_j, u^{{\mathrm{NO2}}\max}_j, u^{{\mathrm{NH4}}\max}_j )\]

with

\[u^{{\mathrm{N..}}\max}_j = V^{{\mathrm{N..}}\max}_j \gamma^{\op{N..}}_j f^{\op{up}}_j(T)\]

Part of the excess carbon is exuded to become DOC,

\[E^{\op{C}}_j = (P^{\op{C}}_j - \mu_j) f^{\op{exude}}_j c_j\]

and part stored as fat,

\[U^{\op{fat}}_j = (P^{\op{C}}_j - \mu_j) (1 - f^{\op{exude}}_j) c_j \;.\]

The Chlorophyll synthesis rate is based on the growth rate (as before).

Table 8.30 Exudation parameters

Trait	Param	Symbol	Default	Units	Description
FracExudeC	a_FracExudeC	\(f^{\op{exude}}_j\)	0.3		fraction of excess carbon exuded