State variables and sub-modules in ICM¶
Core Module
1 PB1 : Diatom g/m^3
2 PB2 : Green Algae g/m^3
3 PB3 : Cyanobacteria g/m^3
4 RPOC : Refractory Particulate Organic Carbon g/m^3
5 LPOC : Labile Particulate Organic Carbon g/m^3
6 DOC : Dissolved Orgnaic Carbon g/m^3
7 RPON : Refractory Particulate Organic Nitrogen g/m^3
8 LPON : Labile Particulate Organic Nitrogen g/m^3
9 DON : Dissolved Orgnaic Nitrogen g/m^3
10 NH4 : Ammonium Nitrogen g/m^3
11 NO3 : Nitrate Nitrogen g/m^3
12 RPOP : Refractory Particulate Organic Phosphorus g/m^3
13 LPOP : Labile Particulate Organic Phosphorus g/m^3
14 DOP : Dissolved Orgnaic Phosphorus g/m^3
15 PO4 : Total Phosphate g/m^3
16 COD : Chemical Oxygen Demand g/m^3
17 DOX : Dissolved Oxygen g/m^3
Silica Module
1 SU : Particulate Biogenic Silica g/m^3
2 SA : Available Silica g/m^3
Zooplankton Module
1 ZB1 : 1st zooplankton g/m^3
2 ZB2 : 2nd zooplankton g/m^3
pH Module
1 TIC : Total Inorganic Carbon g/m^3
2 ALK : Alkalinity g[CaCO3]/m^3
3 CA : Dissolved Calcium g[CaCO3]/m^3
4 CACO3 : Calcium Carbonate g[CaCO3]/m^3
CBP Module
1 SRPOC : Slow Refractory Particulate Organic Carbon g/m^3
2 SRPON : Slow Refractory Particulate Organic Nitro. g/m^3
3 SRPOP : Slow Refractory Particulate Organic Phosp. g/m^3
4 PIP : Particulate Inorganic Phosphate g/m^3
SAV Module (no transport variables)
VEG Module (no transport variables)
SFM Module (no transport variables)
1. Core Module¶
1.1 Mass Balance Equations of State Variables in ICM¶
**Note: **
a). settling of variables are addressed in separate section, and not included in the equations below
b). When iCBP=0, equations of (\(\text{SRPOC},\text{SRPON},\text{SRPOP},\text{PIP}\)) are omiited, and \((FCP_4^m,FCM_4^m,FNP_5^m,FNM_5^m,FPP_5^m,FPM_5^m)=0, (KC_S,KN_S,KP_S,KP_P)=0\)
Phytoplankton (PB1, PB2, PB3):
\[\begin{flalign}
& d\text{PB}^i = \text{GP}^i-\text{MT}^i-\text{PR}^i \text{ , i=1,3} \\
\end{flalign}\]
Carbon (RPOC, LPOC, DOC):
\[\begin{flalign}
& d\text{RPOC} = -KC_1 \cdot \text{RPOC}+ \sum_{m=1,3} \left( FCP_1^m \cdot \text{PR}^m + FCM_1^m \cdot \text{MT}^m \right)\\
& d\text{LPOC} = -KC_2 \cdot \text{LPOC}+ \sum_{m=1,3} \left( FCP_2^m \cdot \text{PR}^m + FCM_2^m \cdot \text{MT}^m \right) \\
& \begin{split}
& d\text{DOC}=KC_1 \cdot \text{RPOC}+ KC_2 \cdot \text{LPOC} + KC_S \cdot \text{SRPOC} -K_{HR} \cdot \text{DOC} -Denit \cdot \text{DOC} \\
& + \sum_{m=1,3} FCP_3^m \cdot \text{PR}^m + \sum_{m=1,3} \left[FCM_3^m+ \left(1-\sum_{i=1,4} FCM_i^m \right) \cdot \frac{KhDO^m}{DO+KhDO^m} \right] \cdot \text{MT}^m \\
& \end{split} \\
& d\text{SRPOC} = -KC_S \cdot \text{SRPOC}+ \sum_{m=1,3} \left( FCP_4^m \cdot \text{PR}^m + FCM_4^m \cdot \text{MT}^m \right)\\
\end{flalign}\]
Nitrogen (RPON, LPON, DON, NH4, NO3):
\[\begin{flalign}
& d\text{RPON} = -KN_1 \cdot \text{RPON}+ \sum_{m=1,3} n2c^m \cdot \left( FNP_1^m \cdot \text{PR}^m + FNM_1^m \cdot \text{MT}^m \right) \\
& d\text{LPON} = -KN_2 \cdot \text{LPON}+ \sum_{m=1,3} n2c^m \cdot \left( FNP_2^m \cdot \text{PR}^m + FNM_2^m \cdot \text{MT}^m \right) \\
& \begin{split}
& d\text{DON} = KN_1 \cdot \text{RPON} + KN_2 \cdot \text{LPON} + KN_S \cdot \text{SRPON} -KN_3 \cdot \text{DON} \\
& + \sum_{m=1,3} n2c^m \cdot \left( FNP_3^m \cdot \text{PR}^m + FNM_3^m \cdot \text{MT}^m \right)
& \end{split} \\
& d\text{NH4} = KN_3 \cdot \text{DON}-Nit \cdot \text{NH4}+ \sum_{m=1,3} n2c^m \cdot \left( FNP_4^m \cdot \text{PR}^m + FNM_4^m \cdot \text{MT}^m -fPN^m \cdot \text{GP}^m \right) \\
& d\text{NO3} = Nit \cdot \text{NH4}-dn2c \cdot Denit \cdot \text{DOC}-\sum_{m=1,3} n2c^m \cdot \left( 1-fPN^m \right) \cdot \text{GP}^m \\
& d\text{SRPON} = -KN_S \cdot \text{SRPON}+ \sum_{m=1,3} n2c^m \cdot \left( FNP_5^m \cdot \text{PR}^m + FNM_5^m \cdot \text{MT}^m \right)\\
\end{flalign}\]
Phosphorus (RPOP, LPOP, DOP, PO4):
\[\begin{flalign}
& d\text{RPOP}= -KP_1 \cdot \text{RPOP}+ \sum_{m=1,3} p2c^m \cdot \left( FPP_1^m \cdot \text{PR}^m + FPM_1^m \cdot \text{MT}^m \right) \\
& d\text{LPOP}= -KP_2 \cdot \text{LPOP}+ \sum_{m=1,3} p2c^m \cdot \left( FPP_2^m \cdot \text{PR}^m + FPM_2^m \cdot \text{MT}^m \right) \\
& \begin{split}
& d\text{DOP} = KP_1 \cdot \text{RPOP} + KP_2 \cdot \text{LPOP} + KP_S \cdot \text{SRPOP} -KP_3 \cdot \text{DOP} \\
& + \sum_{m=1,3} p2c^m \cdot \left( FPP_3^m \cdot \text{PR}^m + FPM_3^m \cdot \text{MT}^m \right)
& \end{split} \\
& d\text{PO4} = KP_3 \cdot \text{DOP} + KP_P \cdot \text{PIP} + \sum_{m=1,3} p2c^m \cdot \left( FPP_4^m \cdot \text{PR}^m + FPM_4^m \cdot \text{MT}^m - \text{GP}^m \right) \\
& d\text{SRPOP} = -KP_S \cdot \text{SRPOP}+ \sum_{m=1,3} p2c^m \cdot \left( FPP_5^m \cdot \text{PR}^m + FPM_5^m \cdot \text{MT}^m \right)\\
& d\text{PIP} = -KP_P \cdot \text{PIP} \\
\end{flalign}\]
Oxygen (COD, DO):
\[\begin{flalign}
& d\text{COD} = -K_{COD} \cdot \text{COD} \\
& \begin{split} \\
& d\text{DO} = -o2n \cdot Nit \cdot \text{NH4} -o2c \cdot K_{HR} \cdot \text{DOC} -K_{COD} \cdot \text{COD} \\
& + \sum_{m=1,3} o2c \cdot \left[ \left(1.3-0.3 \cdot fPN^m \right) \cdot \text{GP}^m -\left(1- \sum_{i=1,4} FCM_i^m \right) \cdot \frac{DO}{DO+KhDO^m} \cdot \text{MT}^m \right] \\
& \end{split} \\
\end{flalign}\]
1.2. Pre-Calculation¶
1.2.1. Growth, metabolism, predation¶
\[\begin{flalign}
& \text{GP}^i = \text{GPM}^i \cdot f(T) \cdot f(Sal) \cdot f(I) \cdot \text{min} \left[ f(N),f(P),f(S) \right] \cdot \text{PB}^i \\
& \text{MT}^i = \text{MTR}^i \cdot \text{GP} +\text{MTB}^i \cdot \text{exp} \left[ KT_{MT}^i \cdot \left( T-T_{MT}^i \right) \right] \cdot \text{PB}^i \\
& \text{PR}^i =
\begin{cases}
& \text{PRR}^i \cdot \text{exp} \left[ KT_{MT}^i \cdot \left( T-T_{MT}^i \right) \right] \cdot \text{PB}^i \text{, iPR=0} \\
& \text{PRR}^i \cdot \text{exp} \left[ KT_{MT}^i \cdot \left( T-T_{MT}^i \right) \right] \cdot \left( \text{PB}^i \right)^2 \text{, iPR=1} \\
\end{cases} \\
\end{flalign}\]
\[\begin{flalign}
& f(I)= \frac{\text{mLight}}{\sqrt{\text{mLight}^2+IK^2}} \\
& f(T)=
\begin{cases}
\text{exp}\left[-KTGP_1^i \cdot \left( T-TGP^i \right)^2 \right] \text{, if }T < TGP^i \\
\text{exp}\left[-KTGP_2^i \cdot \left( T-TGP^i \right)^2 \right] \text{, if }T \geq TGP^i \\
\end{cases} \\
& f(N)= \frac{\text{DIN}}{\text{DIN}+KhN^i} \\
& f(P)= \frac{\text{PO4d}}{\text{PO4d}+KhP^i} \\
& f(Sal)= \frac{KhSal_i^2}{KhSal_i^2+Sal^2} \\
\end{flalign}\]
1.2.2. Decay rates of orgnaic matter¶
\[\begin{flalign}
& KC_i = \left( KC_i^0+KC_i^{alg} \cdot APB \right) \cdot KT_M \\
& KN_i = \left( KN_i^0+KN_i^{alg} \cdot APB \cdot \frac{mKhN}{mKhN+DIN} \right) \cdot KT_M \\
& KP_i = \left( KP_i^0+KP_i^{alg} \cdot APB \cdot \frac{mKhP}{mKhP+PO4_d} \right) \cdot KT_M \\
\end{flalign}\]
\[\begin{flalign}
KT_M=\text{exp} \left[ KT_{RM}^i \cdot \left( T-T_{RM}^i \right) \right]
\end{flalign}\]
1.2.3. Respiration, denitrification, decay of COD, nitrification¶
\[\begin{flalign}
& K_{HR} = KC_3 \cdot \frac{DO}{KhDO_{OX}+DO} \\
& K_{COD}= \frac{DO}{KhCOD+DO} \cdot KCD \cdot \text{exp}[KT_{RCOD} \cdot (T-T_{RCOD})] \\
& Denit = an2c \cdot KC_3 \cdot \frac{KhDO_{OX}}{KhDO_{OX}+DO} \cdot \frac{NO3}{KhNO3_{dn}+NO3} \\
& Nit = Nit^{max} \cdot \frac{DO}{KhDO_n+DO} \cdot \frac{KhNH4_n}{KhNH4_n+NH4} \cdot
\begin{cases}
\text{exp}[-KT_{Nit}^1 \cdot (T_{Nit}-T)] \text{, if } T<T_{Nit} \\
\text{exp}[-KT_{Nit}^2 \cdot (T-T_{Nit})] \text{, if } T \geq T_{Nit}\\
\end{cases}
\end{flalign}\]