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

8.7.3.6. Nutrient uptake and limitation

The uptake rate of dissolved inorganic carbon is

\[U^{\op{DIC}}_j = P^{\mathrm{C}}_j {c}_j - \op{synthcost}\cdot U^{\mathrm{N}}_j\]

where the carbon specific growth rate, \(P^{\op{C}}_j\), is discussed in Section 8.7.3.2, and the second term is only present with both N and Chl quotas and the Geider formulation of photosynthesis.

Nutrient limitation is computed following Liebig’s law of the minimum,

(8.58)\[\gamma^{\op{nut}}_j = \min(\gamma^{\mathrm{P}}_j, \gamma^{\mathrm{N}}_j, \gamma^{\op{Si}}_j, \gamma^{\op{Fe}}_j)\]

We will discuss the limitation terms for each element together with the uptake rate of that element for the cases with and without a corresponding elemental quota in plankton.

8.7.3.6.1. Without P quota:

Monod limitation

\[\gamma^{\mathrm{P}}_j = \frac{\mathrm{PO}_4}{\mathrm{PO}_4 + k^{\op{PO4}}_j}\]

\[U^{\mathrm{P}}_j = R^{{\mathrm{P}}:{\mathrm{C}}}_j U^{\op{DIC}}_j\]

8.7.3.6.2. With P quota:

normalized Droop limitation

\[\gamma^{\mathrm{P}}_j = \left[ \frac{1 - Q^{{\mathrm{P}}\min}_j/Q^{{\mathrm{P}}}_j} {1 - Q^{{\mathrm{P}}\min}_j/Q^{{\mathrm{P}}\max}_j} \right]_0^1\]

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

where

\[{{\text{reg}}}^{Q{\mathrm{P}}}_j = \left( \left[ \frac{Q^{{\mathrm{P}}\max}_j - Q^{{\mathrm{P}}}_j} {Q^{{\mathrm{P}}\max}_j - Q^{{\mathrm{P}}\min}_j} \right]_0^1 \right)^{h_{\op{U}}}\]

and the exponent, \(h_{\op{U}}\), is the Hill number for uptake (default 1).

8.7.3.6.3. Si:

Diatoms (trait hasSi = 1) have linear limitation when using a Si quota,

\[\gamma^{\op{Si}}_j = \left[ \frac{Q^{\op{Si}}_j - Q^{\op{Si}\min}_j} {Q^{\op{Si}\max}_j - Q^{\op{Si}\min}_j} \right]_0^1\]

Otherwise Si is analogous to P.

8.7.3.6.4. Without N quota:

diazotroph:

No limitation, no consumption:

\[\gamma^{\mathrm{N}}_j = 1\]

\[U^{\op{NH4}}_j = U^{\op{NO2}}_j = U^{\op{NO3}}_j = 0\]

not diazotroph:

Modified Monod limitation:

\[\gamma^{\mathrm{N}}_j = \left[ \gamma^{\op{NH4}}_j + \gamma^{\op{NO2}}_j + \gamma^{\op{NO3}}_j \right]_0^1\]

\[\gamma^{\op{NH4}}_j = \op{useNH4}_j \frac{\op{NH}_4}{\op{NH}_4 + k^{\op{NH4}}_j}\]

NO₂ and NO₃ limitations can be combined (trait combNO = 1),

\[\gamma^{\op{NO2}}_j = \op{useNO2}_j \dfrac{\op{NO}_2}{\op{NO}_2 + \op{NO}_3 + k^{\op{NO3}}_j} \op{e}^{-\sigma^{\op{amm}}_j \op{NH}_4}\]

\[\gamma^{\op{NO3}}_j = \op{useNO3}_j \dfrac{\op{NO}_3}{\op{NO}_2 + \op{NO}_3 + k^{\op{NO3}}_j} \op{e}^{-\sigma^{\op{amm}}_j \op{NH}_4}\]

or not (combNO = 0),

\[\gamma^{\op{NO2}}_j = \op{useNO2}_j \dfrac{\op{NO}_2}{\op{NO}_2 + k^{\op{NO2}}_j} \op{e}^{-\sigma^{\op{amm}}_j \op{NH}_4}\]

\[\gamma^{\op{NO3}}_j = \op{useNO3}_j \dfrac{\op{NO}_3}{\op{NO}_3 + k^{\op{NO3}}_j} \op{e}^{-\sigma^{\op{amm}}_j \op{NH}_4}\]

Uptake is then

(8.59)\[ \begin{align}\begin{aligned} U^{\op{NH4}}_j &= \frac{\gamma^{\op{NH4}}_j} {\gamma^{\op{NH4}}_j + \gamma^{\op{NO2}}_j + \gamma^{\op{NO3}}_j} R^{{\mathrm{N}}:{\mathrm{C}}}_j U^{\op{DIC}}_j\\ U^{\op{NO2}}_j &= \frac{\gamma^{\op{NO2}}_j} {\gamma^{\op{NH4}}_j + \gamma^{\op{NO2}}_j + \gamma^{\op{NO3}}_j} R^{{\mathrm{N}}:{\mathrm{C}}}_j U^{\op{DIC}}_j\\ U^{\op{NO3}}_j &= \frac{\gamma^{\op{NO3}}_j} {\gamma^{\op{NH4}}_j + \gamma^{\op{NO2}}_j + \gamma^{\op{NO3}}_j} R^{{\mathrm{N}}:{\mathrm{C}}}_j U^{\op{DIC}}_j\end{aligned}\end{align} \]

8.7.3.6.5. With N quota:

linear limitation

\[\gamma^{\mathrm{N}}_j = \left[ \frac{Q^{{\mathrm{N}}}_j - Q^{{\mathrm{N}}\min}_j} {Q^{{\mathrm{N}}\max}_j - Q^{{\mathrm{N}}\min}_j} \right]_0^1\]

\[ \begin{align}\begin{aligned}U^{\op{NH4}}_j &= V^{\op{NH4}\max}_j \frac{\op{NH}_4}{\op{NH}_4 + k^{\op{NH4}}_j} {{\text{reg}}}^{Q{\mathrm{N}}}_j \cdot f^{{\text{up}}}_j(T) \cdot {c}_j\\U^{\op{NO2}}_j &= V^{\op{NO2}\max}_j \cdot {\mathrm{e}}^{-\sigma^{\op{amm}}_j \op{NH}_4} \cdot \frac{\op{NO}_2}{\op{NO}_2 + k^{\op{NO2}}_j} {{\text{reg}}}^{Q{\mathrm{N}}}_j \cdot f^{{\text{up}}}_j(T) \cdot {c}_j\\U^{\op{NO3}}_j &= V^{\op{NO3}\max}_j \cdot {\mathrm{e}}^{-\sigma^{\op{amm}}_j \op{NH}_4} \cdot \frac{\op{NO}_3}{\op{NO}_3 + k^{\op{NO3}}_j} {{\text{reg}}}^{Q{\mathrm{N}}}_j \cdot f^{{\text{up}}}_j(T) \cdot {c}_j \cdot \gamma^{\op{QFe}}_j\end{aligned}\end{align} \]

where

\[{{\text{reg}}}^{Q{\mathrm{N}}}_j = \left( \left[ \frac{Q^{{\mathrm{N}}\max}_j - Q^{{\mathrm{N}}}_j} {Q^{{\mathrm{N}}\max}_j - Q^{{\mathrm{N}}\min}_j} \right]_0^1 \right)^{h_{\op{U}}}\]

diazotroph:

consume what is available, fix what is missing (up to \(V^{{\mathrm{N}}\max}_j\)),

\[\begin{split}U^{{\mathrm{N}}}_j = \max\biggl( U^{\op{NH4}}_j + U^{\op{NO2}}_j + U^{\op{NO3}}_j,\; V^{{\mathrm{N}}\max}_j {{\text{reg}}}^{Q{\mathrm{N}}}_j \cdot f^{{\text{up}}}_j(T) \cdot {c}_j \biggr) \\\end{split}\]

Rate of nitrogen fixation is

\[U^{{\mathrm{N}}}_j - U^{\op{NH4}}_j - U^{\op{NO2}}_j - U^{\op{NO3}}_j\]

not diazotroph:

\[U^{{\mathrm{N}}}_j = U^{\op{NH4}}_j + U^{\op{NO2}}_j + U^{\op{NO3}}_j\]

8.7.3.6.6. Without Fe quota:

\[\gamma^{\op{Fe}}_j = \frac{\op{FeT}}{\op{FeT}+ k^{\op{Fe}}_j}\]

\[\gamma^{\op{QFe}}_j = 1\]

\[U^{\op{Fe}}_j = R^{\op{Fe}:{\mathrm{C}}}_j U^{\op{DIC}}_j\]

8.7.3.6.7. With Fe quota,

a low iron quota does not directly limit growth,

\[\gamma^{\op{Fe}}_j = 1\]

It rather reduces the light available for photosynthesis (see Section 8.7.3.2 above),

\[\gamma^{\op{QFe}}_j = \left[ \frac{1 - Q^{\op{Fe}\min}_j/Q^{\op{Fe}}_j} {1 - Q^{\op{Fe}\min}_j/Q^{\op{Fe}\max}_j} \right]_0^1\]

Iron uptake depends on the available dissolved iron,

\[U^{\op{Fe}}_j = V^{\op{Fe}\max}_j \frac{\op{FeT}}{\op{FeT}+ k^{\op{Fe}}_j} {{\text{reg}}}^{Q\op{Fe}}_j \cdot f^{{\text{up}}}_j(T) \cdot {c}_j\]

where

\[{{\text{reg}}}^{Q\op{Fe}}_j = \left( \left[ \frac{Q^{\op{Fe}\max}_j - Q^{\op{Fe}}_j} {Q^{\op{Fe}\max}_j - Q^{\op{Fe}\min}_j} \right]_0^1 \right)^{h_{\op{U}}}\]

8.7.3.6.8. Effective half saturation constants

If DARWIN_effective_ksat is true, half saturations for non-quota elements are computed from quota traits. If darwin_select_kn_allom=1 (now deprecated), the half saturation for \(\op{NO}_3\) is computed following Ward et al.,

\[k^{\op{NO3}}_j \rightarrow \frac{ k^{\op{NO3}}_j P^{{\mathrm{C}}{\op{m}}}_j Q^{{\mathrm{N}}\min}_j (Q^{{\mathrm{N}}\max}_j - Q^{{\mathrm{N}}\min}_j) } { V^{\op{NO3}\max}_j Q^{{\mathrm{N}}\max}_j + P^{{\mathrm{C}}{\op{m}}}_j Q^{{\mathrm{N}}\min}_j (Q^{{\mathrm{N}}\max}_j - Q^{{\mathrm{N}}\min}_j) }\]

and those of the other elements are computed by scaling \(k^{\op{NO3}}_j\) with the type’s elemental ratios. Here, \(k^{\op{NO3}}_j\) on the right-hand side is computed from a_ksatNO3 and b_ksatNO3.

If darwin_select_kn_allom=2 (the default), the half saturation for \(\op{NO}_3\) is computed following Follett et al.,

\[k^{\op{NO3}}_j \rightarrow k^{\op{NO3}}_j \frac { P^{{\mathrm{C}}{\op{m}}}_j Q^{{\mathrm{N}}\min}_j } { V^{\op{NO3}\max}_j }\]

Those of the other elements are again computed by scaling \(k^{\op{NO3}}_j\) with the type’s elemental ratios.

8.7.3.6.9. Uptake and limitation parameters

Table 8.36 Uptake parameters

Trait	Param	Symbol	Default	Units	Description
	synthcost	synthcost	0.0	mmol C / mmol N	cost of biosynthesis
hasSi	grp_hasSi	hasSi\(_j\)	0		1: uses silica (Diatom), 0: not
diazo	grp_diazo	diazo\(_j\)	0		1: use molecular instead of mineral nitrogen, 0: not
useNH4	grp_useNH4	useNH4\(_j\)	1		1: can use ammonia, 0: not
useNO2	grp_useNO2	useNO2\(_j\)	1		1: can use nitrite, 0: not
useNO3	grp_useNO3	useNO3\(_j\)	1		1: can use nitrate, 0: not
combNO	grp_combNO	combNO\(_j\)	1		1: combined nitrite/nitrate limitation, 0: not
Qnmin	a,b_Qnmin	\(Q^{\op{N}\min}_j\)	0.07 V–0.17	mmol N / mmol C	minimum nitrogen quota
Qnmax	a,b_Qnmax	\(Q^{\op{N}\op{max}}_j\)	0.25 V–0.13	mmol N / mmol C	maximum nitrogen quota
Qpmin	a,b_Qpmin	\(Q^{\op{P}\min}_j\)	0.002 V0	mmol P / mmol C	minimum phosphorus quota
Qpmax	a,b_Qpmax	\(Q^{\op{P}\op{max}}_j\)	0.01 V0	mmol P / mmol C	maximum phosphorus quota
Qsimin	a,b_Qsimin	\(Q^{\op{Si}\min}_j\)	0.002 V0	mmol Si / mmol C	minimum silica quota
Qsimax	a,b_Qsimax	\(Q^{\op{Si}\op{max}}_j\)	0.004 V0	mmol Si / mmol C	maximum silica quota
Qfemin	a,b_Qfemin	\(Q^{\op{Fe}\min}_j\)	15E-6 V0	mmol Fe / mmol C	minimum iron quota
Qfemax	a,b_Qfemax	\(Q^{\op{Fe}\op{max}}_j\)	80E-6 V0	mmol Fe / mmol C	maximum iron quota
vmaxNO3	a,b_vmaxNO3	\(V^{\op{NO3}\op{max}}_j\)	(0.26/day) V–0.27	mmol N / (mmol C s)	maximum nitrate uptake rate
vmaxNO2	a,b_vmaxNO2	\(V^{\op{NO2}\op{max}}_j\)	(0.51/day) V–0.27	mmol N / (mmol C s)	maximum nitrite uptake rate
vmaxNH4	a,b_vmaxNH4	\(V^{\op{NH4}\op{max}}_j\)	(0.51/day) V–0.27	mmol N / (mmol C s)	maximum ammonia uptake rate
vmaxN	a,b_vmaxN	\(V^{\op{N}\op{max}}_j\)	(1.28/day) V–0.27	mmol N / (mmol C s)	maximum nitrogen uptake rate for diazotrophs
vmaxPO4	a,b_vmaxPO4	\(V^{\op{PO4}\op{max}}_j\)	(0.077/day) V–0.27	mmol P / (mmol C s)	maximum phosphate uptake rate
vmaxSiO2	a,b_vmaxSiO2	\(V^{\op{SiO2}\op{max}}_j\)	(0.077/day) V–0.27	mmol Si / (mmol C s)	maximum silica uptake rate
vmaxFeT	a,b_vmaxFeT	\(V^{\op{Fe}\op{max}}_j\)	(14E-6/day) V–0.27	mmol Fe / (mmol C s)	maximum iron uptake rate
ksatNO3	a,b_ksatNO3	\(k^{\op{NO3}}_j\)	0.085 V0.27	mmol N m-3	half-saturation conc. for nitrate uptake/limitation
ksatNO2	a,b_ksatNO2	\(k^{\op{NO2}}_j\)	0.17 V0.27	mmol N m-3	half-saturation conc. for nitrite uptake/limitation
ksatNH4	a,b_ksatNH4	\(k^{\op{NH4}}_j\)	0.17 V0.27	mmol N m-3	half-saturation conc. for ammonia uptake/limitation
ksatPO4	a,b_ksatPO4	\(k^{\op{PO4}}_j\)	0.026 V0.27	mmol P m-3	half-saturation conc. for phosphate uptake/limitation
ksatSiO2	a,b_ksatSiO2	\(k^{\op{SiO2}}_j\)	0.024 V0.27	mmol Si m-3	half-saturation conc. for silica uptake/limitation
ksatFeT	a,b_ksatFeT	\(k^{\op{Fe}}_j\)	80E-6 V0.27	mmol Fe m-3	half-saturation conc. for iron uptake/limitation
	a_ksatNO2fac		1		used for eff.ksat
	a_ksatNH4fac		0.5		used for eff.ksat
R_NC	a_R_NC	\(R^{\op{N}:\op{C}}_j\)	16/120	mmol N / mmol C	nitrogen-carbon ratio
R_PC	a_R_PC	\(R^{\op{P}:\op{C}}_j\)	1/120	mmol P / mmol C	phosphorus-carbon ratio
R_SiC	a_R_SiC	\(R^{\op{Si}:\op{C}}_j\)	0	mmol Si / mmol C	silica-carbon ratio
R_FeC	a_R_FeC	\(R^{\op{Fe}:\op{C}}_j\)	1E-3/120	mmol Fe / mmol C	iron-carbon ratio
R_ChlC	a_R_ChlC	\(R^{\op{chl}c}_j\)	16/120	mg Chl / mmol C	chlorophyll-carbon ratio
amminhib	a_amminhib	\(\sigma^{\op{amm}}_j\)	4.6	m3 / mmol N	coefficient for NH4 inhibition of NO uptake
	hillnumUptake	\(h^{\op{U}}\)	1.0		exponent for limiting quota uptake in nutrient uptake