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

8.7.3.1. Model equations

The basic model equations are (omitting transport terms handled by the ptracers package, Sinking and Swimming, Air-sea exchanges and terms correcting conservation with the linear free surface formulation discussed in Section 8.7.3.24):

\[\begin{split}\partial_t\op{DIC} &= \sum_j \bigl( (-U^{\mathrm{DIC}}_j + R^{\mathrm{C}}_j) \cdot (1 + R^{\text{PIC:POC}}_j) + R^{\mathrm{DIC}}_j \bigr) + R_{\mathrm{DOC}} + [R_{\mathrm{POC}}] + D_{\mathrm{PIC}} \\ \partial_t\op{PO}_4 &= \sum_j \bigl( -U^{\mathrm{PO4}}_j + R^{\mathrm{P}}_j + R^{\mathrm{PO4}}_j \bigr) + R_{\mathrm{DOP}} + [R_{\mathrm{POP}}] \\ \partial_t\op{NH}_4 &= \sum_j \bigl( -U^{\mathrm{NH4}}_j + R^{\mathrm{N,NH4}}_j + R^{\mathrm{NH4}}_j \bigr) + R_{\mathrm{DON}} + [R_{\mathrm{PON}}] - P_{\mathrm{NO2}} - D_{\mathrm{NH4}} \\ \partial_t\op{NO}_2 &= \sum_j \bigl( -U^{\mathrm{NO2}}_j + R^{\mathrm{N,NO2}}_j + R^{\mathrm{NO2}}_j \bigr) + P_{\mathrm{NO2}} - P_{\mathrm{NO3}} \\ \partial_t\op{NO}_3 &= \sum_j \bigl( -U^{\mathrm{NO3}}_j + R^{\mathrm{N,NO3}}_j + R^{\mathrm{NO3}}_j \bigr) + P_{\mathrm{NO3}} - D_{\mathrm{NO3}} \\ \partial_t\op{FeT} &= \sum_j \bigl( -U^{\mathrm{Fe}}_j + R^{\mathrm{Fe}}_j + R^{\mathrm{FeT}}_j \bigr) + R_{\mathrm{DOFe}} + [R_{\mathrm{POFe}}] + S_{\mathrm{Fe}} \\ \partial_t\op{SiO}_2 &= \sum_j \bigl( -U^{\mathrm{SiO2}}_j + R^{\mathrm{Si}}_j \bigr) + R_{\mathrm{POSi}} \\ \partial_t c_j &= U^{\mathrm{DIC}}_j - M_j - R^{\mathrm{C}}_j - G_j + g_j^{\mathrm{C}} \\ \partial_t p_j &= U^{\mathrm{PO4}}_j - M_j Q^{\mathrm{P}}_j - G_j Q^{\mathrm{P}}_j + g_j^{\mathrm{P}} \qquad\text{(with P quota)} \\ \partial_t n_j &= U^{{\mathrm{N}}}_j \;\;\; - M_j Q^{\mathrm{N}}_j - G_j Q^{\mathrm{N}}_j + g_j^{\mathrm{N}} \qquad\text{(with N quota)} \\ \partial_t\op{fe}_j &= U^{\mathrm{Fe}}_j \;\; - M_j Q^{\mathrm{Fe}}_j - G_j Q^{\mathrm{Fe}}_j + g_j^{\mathrm{Fe}} \qquad\text{(with Fe quota)} \\ \partial_t\op{si}_j &= U^{\mathrm{SiO2}}_j - M_j Q^{\mathrm{Si}}_j - G_j Q^{\mathrm{Si}}_j \qquad\text{(with Si quota)} \\ \partial_t\op{Chl}_j &= S^{\mathrm{Chl}}_j \; - M_j Q^{\mathrm{Chl}}_j - G_j Q^{\mathrm{Chl}}_j \qquad\text{(with Chl quota)} \\ \partial_t\op{DOC} &= \sum_j M_j^{\mathrm{DOM}} \;\;\;\;\; + g^{\mathrm{DOC}} + \sum_j \left( H^{\mathrm{POC}}_j - U^{\mathrm{DOC}}_j \right) - R_{\mathrm{DOC}} - S_{\mathrm{CDOM}} R^{{\mathrm{C}}:{\mathrm{P}}}_{\mathrm{CDOM}} \\ \partial_t\op{DOP} &= \sum_j M_j^{\mathrm{DOM}} Q_j^{{\mathrm{P}}} + g^{\mathrm{DOP}} + \sum_j \left( H^{\mathrm{POP}}_j - U^{\mathrm{DOP}}_j \right) - R_{\mathrm{DOP}} - S_{\mathrm{CDOM}} \\ \partial_t\op{DON} &= \sum_j M_j^{\mathrm{DOM}} Q_j^{{\mathrm{N}}} + g^{\mathrm{DON}} + \sum_j \left( H^{\mathrm{PON}}_j - U^{\mathrm{DON}}_j \right) - R_{\mathrm{DON}} - S_{\mathrm{CDOM}} R^{{\mathrm{N}}:{\mathrm{P}}}_{\mathrm{CDOM}} \\ \partial_t\op{DOFe} &= \sum_j M_j^{\mathrm{DOM}} Q_j^{\mathrm{Fe}} + g^{\mathrm{DOFe}} + \sum_j \left( H^{\mathrm{POFe}}_j- U^{\mathrm{DOFe}}_j\right) - R_{\mathrm{DOFe}}- S_{\mathrm{CDOM}} R^{{\mathrm{Fe}}:{\mathrm{P}}}_{\mathrm{CDOM}} \\ \partial_t\op{PIC} &= \sum_j M_j R_j^{\text{PIC:POC}} + g^{\mathrm{PIC}} - D_{\mathrm{PIC}} \\ \partial_t\op{POC} &= \sum_j M_j^{\mathrm{POM}} \;\;\;\;\; + g^{\mathrm{POC}} - \sum_j U^{\mathrm{POC}}_j - R_{\mathrm{POC}} \\ \partial_t\op{POP} &= \sum_j M_j^{\mathrm{POM}} Q_j^{{\mathrm{P}}} + g^{\mathrm{POP}} - \sum_j U^{\mathrm{POP}}_j - R_{\mathrm{POP}} \\ \partial_t\op{PON} &= \sum_j M_j^{\mathrm{POM}} Q_j^{{\mathrm{N}}} + g^{\mathrm{PON}} - \sum_j U^{\mathrm{PON}}_j - R_{\mathrm{PON}} \\ \partial_t\op{POFe} &= \sum_j M_j^{\mathrm{POM}} Q_j^{\mathrm{Fe}} + g^{\mathrm{POFe}} - \sum_j U^{\mathrm{POFe}}_j - R_{\mathrm{POFe}} \\ \partial_t\op{POSi} &= \sum_j M_j Q_j^{\mathrm{Si}} \;\;\;\;\; + g^{\mathrm{POSi}} - R_{\mathrm{POSi}} \\ \partial_t\op{ALK} &= -\biggl( P_{\mathrm{NO3}} - \sum_j U^{\mathrm{NO3}}_j \biggr) - 2\biggl( \sum_j U^{\mathrm{DIC}}_j R^{{\text{PIC:POC}}}_j - D_{\mathrm{PIC}} \biggr) + D_{\mathrm{NO3}} \\ \partial_t{\mathrm{O}}_2 &= R_{\mathrm{O}_2:\mathrm{P}} \biggl( \sum_j U^{\mathrm{PO4}}_j - R_{\mathrm{DOP}} - [R_{\mathrm{POP}}] \biggr) \\ \partial_t\op{CDOM} &= S_{\mathrm{CDOM}} \qquad\text{(with CDOM tracer)} \\\end{split}\]

The quotas are defined as \(Q^{\mathrm{P}}_j=p_j/c_j\), etc. The following abbreviated source terms are described in sections below:

• \(U\): uptake, see Nutrient uptake and limitation and Bacteria, also Growth, Internal carbon store and exudation, Non-spectral Light and Spectral Light

• \(M\): Mortality

• \(R^{\mathrm{C}}\), \(R^{\mathrm{P}}\), …: Respiration

• \(H\), \(R^{\mathrm{DIC}}\), …: bacterial hydrolysis and remineralization, see Bacteria

• \(R\), \(P\): parameterized Remineralization and Nitrification

• \(D_{\mathrm{NO3}}\): denitrification, see Denitrification

• \(D_{\mathrm{PIC}}\): dissolution of PIC, see Carbon chemistry

• \(g\), \(G\): grazing gains and losses, see Grazing

• \(S^{\mathrm{Chl}}\): synthesis, see Chlorophyll synthesis

• \(S_{\mathrm{Fe}}\): iron sources, see Iron chemistry

• \(S_{\mathrm{CDOM}}\): see Dynamic CDOM.

With DARWIN_ALLOW_CDOM, all particulate remineralization terms (in square brackets […]) except Si are absent. Without DARWIN_ALLOW_CDOM, \(f_{\mathrm{CDOM}}=0\) and there is no CDOM tracer. The Alk and O₂ tracers are only present with DARWIN_ALLOW_CARBON.

Table 8.28 General parameters

Trait	Param	Symbol	Default	Units	Description
R_PICPOC	a_R_PICPOC	\(R^{\text{PIC:POC}}_j\)	0.8	mmol PIC / mmol POC	inorganic-organic carbon ratio
	R_OP	\(R_{\mathrm{O}_2:\mathrm{P}}\)	170	mmol O2 / mmol P	O2:P ratio for respiration and consumption

For CDOM elemental ratios, see Dynamic CDOM.

Flow diagrams for the various chemical elements:

Figure 8.16 The flow of phosphorus without CDOM (for grazing see Figure 8.18)

Figure 8.17 The flow of phosphorus with CDOM.

Figure 8.18 The flow of phosphorus due to grazing.

Figure 8.19 The flow of nitrogen (no CDOM, grazing suppressed).

Figure 8.20 The flow of iron (no CDOM, grazing suppressed).

Figure 8.21 The flow of silica (no CDOM, grazing suppressed).

Figure 8.22 Sources and sinks of alkalinity.

Figure 8.23 Sources and sinks of oxygen.