.. include:: ../defs.hrst .. _Uptake: Nutrient uptake and limitation ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ The uptake rate of dissolved inorganic carbon is .. math:: U^{\op{DIC}}_j = P^{\mathrm{C}}_j {c}_j - \op{synthcost}\cdot U^{\mathrm{N}}_j where the carbon specific growth rate, :math:`P^{\op{C}}_j`, is discussed in :numref:`Growth`, 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, .. math:: :label: eq_darwin_limitnut \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. Without P quota: '''''''''''''''' Monod limitation .. math:: \gamma^{\mathrm{P}}_j = \frac{\mathrm{PO}_4}{\mathrm{PO}_4 + k^{\op{PO4}}_j} .. math:: U^{\mathrm{P}}_j = R^{{\mathrm{P}}:{\mathrm{C}}}_j U^{\op{DIC}}_j With P quota: ''''''''''''' normalized Droop limitation .. math:: \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 .. math:: 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 .. math:: {{\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, :math:`h_{\op{U}}`, is the Hill number for uptake (default 1). Si: ''' Diatoms (trait :varlink:`hasSi` = 1) have linear limitation when using a Si quota, .. math:: \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. Without N quota: '''''''''''''''' diazotroph: """"""""""" No limitation, no consumption: .. math:: \gamma^{\mathrm{N}}_j = 1 .. math:: U^{\op{NH4}}_j = U^{\op{NO2}}_j = U^{\op{NO3}}_j = 0 not diazotroph: """"""""""""""" Modified Monod limitation: .. math:: \gamma^{\mathrm{N}}_j = \left[ \gamma^{\op{NH4}}_j + \gamma^{\op{NO2}}_j + \gamma^{\op{NO3}}_j \right]_0^1 .. math:: \gamma^{\op{NH4}}_j = \op{useNH4}_j \frac{\op{NH}_4}{\op{NH}_4 + k^{\op{NH4}}_j} NO\ :sub:`2` and NO\ :sub:`3` limitations can be combined (trait :varlink:`combNO` = 1), .. math:: \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} .. math:: \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), .. math:: \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} .. math:: \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 .. math:: :label: eq_uptakeN 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 With N quota: ''''''''''''' linear limitation .. math:: \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 .. math:: 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 where .. math:: {{\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 :math:`V^{{\mathrm{N}}\max}_j`), .. math:: 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) \\ Rate of nitrogen fixation is .. math:: U^{{\mathrm{N}}}_j - U^{\op{NH4}}_j - U^{\op{NO2}}_j - U^{\op{NO3}}_j not diazotroph: """"""""""""""" .. math:: U^{{\mathrm{N}}}_j = U^{\op{NH4}}_j + U^{\op{NO2}}_j + U^{\op{NO3}}_j Without Fe quota: ''''''''''''''''' .. math:: \gamma^{\op{Fe}}_j = \frac{\op{FeT}}{\op{FeT}+ k^{\op{Fe}}_j} .. math:: \gamma^{\op{QFe}}_j = 1 .. math:: U^{\op{Fe}}_j = R^{\op{Fe}:{\mathrm{C}}}_j U^{\op{DIC}}_j With Fe quota, '''''''''''''' a low iron quota does not directly limit growth, .. math:: \gamma^{\op{Fe}}_j = 1 It rather reduces the light available for photosynthesis (see :numref:`Growth` above), .. math:: \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, .. math:: 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 .. math:: {{\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}}} Effective half saturation constants ''''''''''''''''''''''''''''''''''' If :varlink:`DARWIN_effective_ksat` is true, half saturations for non-quota elements are computed from quota traits. If :varlink:`darwin_select_kn_allom`\ =1 (now deprecated), the half saturation for :math:`\op{NO}_3` is computed following Ward et al., .. math:: 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 :math:`k^{\op{NO3}}_j` with the type’s elemental ratios. Here, :math:`k^{\op{NO3}}_j` on the right-hand side is computed from :varlink:`a_ksatNO3` and :varlink:`b_ksatNO3`. If :varlink:`darwin_select_kn_allom`\ =2 (the default), the half saturation for :math:`\op{NO}_3` is computed following Follett et al., .. math:: 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 :math:`k^{\op{NO3}}_j` with the type’s elemental ratios. Uptake and limitation parameters '''''''''''''''''''''''''''''''' .. csv-table:: Uptake parameters :delim: & :widths: 13,20,17,15,15,20 :class: longtable :header: Trait, Param, Symbol, Default, Units, Description :name: tab_phys_pkg_darwin_uptake & :varlink:`synthcost` & :varlink:`synthcost` & 0.0 & mmol C / mmol N & cost of biosynthesis :varlink:`hasSi` & :varlink:`grp_hasSi` & hasSi\ :math:`_j` & 0 & & 1: uses silica (Diatom), 0: not :varlink:`diazo` & :varlink:`grp_diazo` & diazo\ :math:`_j` & 0 & & 1: use molecular instead of mineral nitrogen, 0: not :varlink:`useNH4` & :varlink:`grp_useNH4` & useNH4\ :math:`_j` & 1 & & 1: can use ammonia, 0: not :varlink:`useNO2` & :varlink:`grp_useNO2` & useNO2\ :math:`_j` & 1 & & 1: can use nitrite, 0: not :varlink:`useNO3` & :varlink:`grp_useNO3` & useNO3\ :math:`_j` & 1 & & 1: can use nitrate, 0: not :varlink:`combNO` & :varlink:`grp_combNO` & combNO\ :math:`_j` & 1 & & 1: combined nitrite/nitrate limitation, 0: not :varlink:`Qnmin` & :varlink:`a `,\ :varlink:`b_Qnmin` & :math:`Q^{\op{N}\min}_j` & 0.07 V\ :sup:`--0.17` & mmol N / mmol C & minimum nitrogen quota :varlink:`Qnmax` & :varlink:`a `,\ :varlink:`b_Qnmax` & :math:`Q^{\op{N}\op{max}}_j` & 0.25 V\ :sup:`--0.13` & mmol N / mmol C & maximum nitrogen quota :varlink:`Qpmin` & :varlink:`a `,\ :varlink:`b_Qpmin` & :math:`Q^{\op{P}\min}_j` & 0.002 V\ :sup:`0` & mmol P / mmol C & minimum phosphorus quota :varlink:`Qpmax` & :varlink:`a `,\ :varlink:`b_Qpmax` & :math:`Q^{\op{P}\op{max}}_j` & 0.01 V\ :sup:`0` & mmol P / mmol C & maximum phosphorus quota :varlink:`Qsimin` & :varlink:`a `,\ :varlink:`b_Qsimin` & :math:`Q^{\op{Si}\min}_j` & 0.002 V\ :sup:`0` & mmol Si / mmol C & minimum silica quota :varlink:`Qsimax` & :varlink:`a `,\ :varlink:`b_Qsimax` & :math:`Q^{\op{Si}\op{max}}_j` & 0.004 V\ :sup:`0` & mmol Si / mmol C & maximum silica quota :varlink:`Qfemin` & :varlink:`a `,\ :varlink:`b_Qfemin` & :math:`Q^{\op{Fe}\min}_j` & 15E-6 V\ :sup:`0` & mmol Fe / mmol C & minimum iron quota :varlink:`Qfemax` & :varlink:`a `,\ :varlink:`b_Qfemax` & :math:`Q^{\op{Fe}\op{max}}_j` & 80E-6 V\ :sup:`0` & mmol Fe / mmol C & maximum iron quota :varlink:`vmaxNO3` & :varlink:`a `,\ :varlink:`b_vmaxNO3` & :math:`V^{\op{NO3}\op{max}}_j` & (0.26/day) V\ :sup:`--0.27` & mmol N / (mmol C s) & maximum nitrate uptake rate :varlink:`vmaxNO2` & :varlink:`a `,\ :varlink:`b_vmaxNO2` & :math:`V^{\op{NO2}\op{max}}_j` & (0.51/day) V\ :sup:`--0.27` & mmol N / (mmol C s) & maximum nitrite uptake rate :varlink:`vmaxNH4` & :varlink:`a `,\ :varlink:`b_vmaxNH4` & :math:`V^{\op{NH4}\op{max}}_j` & (0.51/day) V\ :sup:`--0.27` & mmol N / (mmol C s) & maximum ammonia uptake rate :varlink:`vmaxN` & :varlink:`a `,\ :varlink:`b_vmaxN` & :math:`V^{\op{N}\op{max}}_j` & (1.28/day) V\ :sup:`--0.27` & mmol N / (mmol C s) & maximum nitrogen uptake rate for diazotrophs :varlink:`vmaxPO4` & :varlink:`a `,\ :varlink:`b_vmaxPO4` & :math:`V^{\op{PO4}\op{max}}_j` & (0.077/day) V\ :sup:`--0.27` & mmol P / (mmol C s) & maximum phosphate uptake rate :varlink:`vmaxSiO2` & :varlink:`a `,\ :varlink:`b_vmaxSiO2` & :math:`V^{\op{SiO2}\op{max}}_j` & (0.077/day) V\ :sup:`--0.27` & mmol Si / (mmol C s) & maximum silica uptake rate :varlink:`vmaxFeT` & :varlink:`a `,\ :varlink:`b_vmaxFeT` & :math:`V^{\op{Fe}\op{max}}_j` & (14E-6/day) V\ :sup:`--0.27` & mmol Fe / (mmol C s) & maximum iron uptake rate :varlink:`ksatNO3` & :varlink:`a `,\ :varlink:`b_ksatNO3` & :math:`k^{\op{NO3}}_j` & 0.085 V\ :sup:`0.27` & mmol N m\ :sup:`-3` & half-saturation conc. for nitrate uptake/limitation :varlink:`ksatNO2` & :varlink:`a `,\ :varlink:`b_ksatNO2` & :math:`k^{\op{NO2}}_j` & 0.17 V\ :sup:`0.27` & mmol N m\ :sup:`-3` & half-saturation conc. for nitrite uptake/limitation :varlink:`ksatNH4` & :varlink:`a `,\ :varlink:`b_ksatNH4` & :math:`k^{\op{NH4}}_j` & 0.17 V\ :sup:`0.27` & mmol N m\ :sup:`-3` & half-saturation conc. for ammonia uptake/limitation :varlink:`ksatPO4` & :varlink:`a `,\ :varlink:`b_ksatPO4` & :math:`k^{\op{PO4}}_j` & 0.026 V\ :sup:`0.27` & mmol P m\ :sup:`-3` & half-saturation conc. for phosphate uptake/limitation :varlink:`ksatSiO2` & :varlink:`a `,\ :varlink:`b_ksatSiO2` & :math:`k^{\op{SiO2}}_j` & 0.024 V\ :sup:`0.27` & mmol Si m\ :sup:`-3` & half-saturation conc. for silica uptake/limitation :varlink:`ksatFeT` & :varlink:`a `,\ :varlink:`b_ksatFeT` & :math:`k^{\op{Fe}}_j` & 80E-6 V\ :sup:`0.27` & mmol Fe m\ :sup:`-3` & half-saturation conc. for iron uptake/limitation & :varlink:`a_ksatNO2fac` & & 1 & & *used for eff.ksat* & :varlink:`a_ksatNH4fac` & & 0.5 & & *used for eff.ksat* :varlink:`R_NC` & :varlink:`a_R_NC` & :math:`R^{\op{N}:\op{C}}_j` & 16/120 & mmol N / mmol C & nitrogen-carbon ratio :varlink:`R_PC` & :varlink:`a_R_PC` & :math:`R^{\op{P}:\op{C}}_j` & 1/120 & mmol P / mmol C & phosphorus-carbon ratio :varlink:`R_SiC` & :varlink:`a_R_SiC` & :math:`R^{\op{Si}:\op{C}}_j` & 0 & mmol Si / mmol C & silica-carbon ratio :varlink:`R_FeC` & :varlink:`a_R_FeC` & :math:`R^{\op{Fe}:\op{C}}_j` & 1E-3/120 & mmol Fe / mmol C & iron-carbon ratio :varlink:`R_ChlC` & :varlink:`a_R_ChlC` & :math:`R^{\op{chl}c}_j` & 16/120 & mg Chl / mmol C & chlorophyll-carbon ratio :varlink:`amminhib` & :varlink:`a_amminhib` & :math:`\sigma^{\op{amm}}_j` & 4.6 & m\ :sup:`3` / mmol N & coefficient for NH4 inhibition of NO uptake & :varlink:`hillnumUptake` & :math:`h^{\op{U}}` & 1.0 & & exponent for limiting quota uptake in nutrient uptake