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The NPZ (Nutrients, Phytoplankton, Zooplankton) model is based on the model that is developed by Carrasco de la Cruz (2018). The model consists of 9 state variables of which 4 in the water column (DIN, Phyto, Zoo, Det) and 5 in the bottom (MUS, COC, OYS, MPB and Bot_Det). All state variables are expressed in terms of nitrogen (N). The state variables in the water column have the unit of mmol N m<sup>-3</sup> and the state variables in the bottom have a unit of mmol N m<sup>-2</sup>. The state variables in the water column can be transported between the compartments and the North Sea by tidal currents. Also they can be introduced into the Oosterschelde through the Krammersluizen. The state variables in the bottom are not exchanged between the compartments.
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# State variables
## Dissolved inorganic nitrogen (DIN)
Dissolved inorganic nutrients (DIN) is the sum of NO<sub>2</sub>, NO<sub>3</sub> and NH<sub>4</sub>. The unit is mmol N m<sup>-3</sup>. DIN is used by phytoplankton and microphytobenthos for primary production (DINuptake and DINupMPB, respectively). DIN is released by mineralization of detritus in the water column (Mineralization) and in the bottom (BotMin) and respiration by zooplankton (ZooExcretion) and shellfish (DINprod).
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## Phytoplankton (Phyto)
Phytoplankton is composed of microalgae in the water column that are transported by tidal currents. The unit of phytoplankton is mmol N m<sup>-3</sup>. Phytoplankton increases by primary production (DINuptake). Phytoplankton is consumed by zooplankton (ZooGrazing) and shellfish (GrazingPhy). Also phytoplankton can sink to the bottom (SinkPhy), where it becomes part of the bottom detritus (BOT_DET).
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## Zooplankton (Zoo)
Zooplankton (Zoo) are small animals (e.g. crustaceans, shellfish larvae) that live in the water column and are transported by the tidal currents. The unit is mmol N m<sup>-3</sup>. Zooplankton consumes phytoplankton (ZooGrazing). A fraction of the nitrogen that is grazed is lost as feaces (ZooFaeces) and the rest will result in an increase of the zooplankton biomass. Zooplankton can also be consumed by shellfish (GrazingZoo). When the zooplankton dies (ZooMortality) it will become detritus (Det). Finally, zooplankton also produces dissolved inorganic nitrogen by respiration (ZooExcretion).
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## Detritus (Det)
Detritus (Det) is the dead organic carbon. The unit is mmol N m<sup>-3</sup>. Detritus is produced by mortality of zooplankton (ZooMortality) and the production of faeces by zooplankton (ZooFaeces). Shellfish can also filter detritus from the water column (GrazingDet). Finally detritus can sink to the seafloor (SinkDet) where it becomes Bot_Det .
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## Microphytobenthos (MPB)
Microphytobenthos (MPD) are microalgae that live on sea floor. The unit is mmol N m<sup>-2</sup>. MPB only occurs on the intertidal areas. The biomass of MPB increases through primary production (DINuptMPB). When the MPB dies (MortMPB) the nitrogen goes to the bottom detritus (Bot_Det).
## Detritus in the bottom (Bot_Det)
Detritus in the bottom (Bot_Det, mmol N m<sup>-2</sup>) increases by the sinking of detritus and phytoplankton (SinkDet and SinkPhy, respectively). When the shellfish (MUSS, COC and OYSS) die, the nitrogen also goes to the pool of Bot_Det (MortShellfish). Finally the faeces and pseudo faeces that is produced by the shellfish is also a source for the Bot_Det (PseudoFaeces). Nitrogen is released to the water column through mineralization of the Bot_Det (BotMin) where it becomes part of the DIN.
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## Shellfish (MUS, OYS and COC)
Three shellfish species are included in the model, mussels (MUS), Pacific oysters (OYS) and cockles (COC). The units are in mmol N m<sup>-2</sup>. The mussels are only present at the mussel culture plots in the Oosterschelde, oysters are present at the oyster culture plots and at the wild oyster beds. Cockles are only present at the intertidal flats. The physiology of the shellfish is described by the function “SFG()”. This function calculates clearance rate (Clearance), filtration rate of phytoplankton, zooplankton and detritus (GrazingPhy, GrazingZoo and GrazingDet, respectively), respiration rate (DINprod), the production of faeces and pseudo faeces (PseudoFaeces) and the growth of the shellfish (dFF).
The shellfish populations are controlled by a variable mortality (MortShellfish) which is a function of the total stock and the carrying capacity.
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# Physical processes
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## Temperature correction factor
Most of the processed are influenced by the temperature. In the model a temperature correction factor, $`T_{fac}`$. [-] is used to correct for ambient water temperature.
```math
T_{fac} = e^{( \frac{T-10}{10}\cdot\log(Q_{10}))}
```
Where T is the ambient water temperature and $`Q_{10}`$ indicates the relative increase in the rate for an increase of the water temperature with 10 &deg;C.
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## SinkDet
The detritus sinks to the sea floor with a constant sinking velocity ($`SinkinRate_{Det}`$)[m d<sup>-1</sup>]. The units of SinkDet is [mmol N m<sup>-2</sup> d<sup>-1</sup>].
```math
SinkDet = SinkinRate_{Det}\cdot\ Det
```
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## SinkPhy
The phytoplankton sinks to the sea floor with a constant sinking velocity ($`SinkinRate_{Phy}`$)[m d<sup>-1</sup>]. The units of SinkPhy is [mmol N m<sup>-2</sup> d<sup>-1</sup>].
```math
SinkPhy = SinkinRate_{Phy}\cdot\ Phyto
```
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## Transport
State variables in the water column (DIN, Phyto, Zoo and Det) are transported by tidal currents. This transport is modeled as exchange rates between the neighboring compartments. It is assumed that the exchange rates are bi-directional, i.e. the exchange rate [m<sup>3</sup> d<sup>-1</sup>]. from compartment 1 to compartment 2 is the same as the exchange rate from compartment 2 to 1.
```math
Flux_{1,2} = \frac{Ex_{1,2}\cdot(C_{1}-C_{2})}{Vol_{2}}
```
Where $`Flux_{1,2}`$ is the import from compartment 2 to compartment 1 [mmol N m<sup>-3</sup> d<sup>-1</sup>]. $`Ex_{1,2}`$ is the exchange rate between compartment 1 and 2 [m<sup>3</sup> d<sup>-1</sup>]. $`C_{1}`$ and $`C_{2}`$ are the concentrations of the state variables in compartment 1 and 2 [mmol N m<sup>-3</sup>], respectively. $`Vol_{2}`$ is the volume of compartment 2 [m<sup>3</sup>].
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# Biological processes
## DINuptake
DINuptake is the production of phytoplankton (primary production) [mmolN m<sup>-3</sup> d<sup>-1</sup>] and is depending on the underwater-light conditions, water temperature, the nutrient concentration (DIN) and the phytoplankton concentration.
Mean daily irradiation at the water surface ($`I_{0}`$) [W d<sup>-1</sup>] is derived from the file KNMI measurements (stored in “./DataOS/KNMI/”). The underwater-light conditions are calculated from the daily irradiation at the water surface and the extinction coefficient ($`k_{d}`$):
```math
I_{z} = I_{0} \cdot e^{-k_{d} \cdot z}
```
The average light ($`I_{avg}`$) is calculated by integration of the function over the whole water column divided over the waterdepth.
```math
I_{avg} = \frac{ \int_{z=0}^{z=Depth} \cdot I_{z} \cdot dz}{Depth}
```
It is assumed that about 50% of the light is photosynthetic active radiation (PAR) $`PAR = 0.5 /cdot I_{avg}`$
DINuptake is calculated from the maximum specific uptake rate [dsup>-1</sup>] at ambient temperature and the algal concentration. Limitations by light and nutrients are described by Monod formulations.
```math
DIN_{upt} = DIN_{upt,max} \cdot T_{fac} \cdot Phyto \cdot (\frac{PAR}{PAR+ks_{PAR}}) \cdot (\frac{DIN}{DIN+ks_{DIN}})
```
Where $`DIN_{upt,max}`$ is the maximum DIN uptake rate of algae at 10&deg;C [d<sup>-1</sup>], $`ks_{PAR}`$ and $`ks_{DIN}`$ are the half-saturation coefficients for light and DIN, respectively.
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## ZooGrazing
The phytoplankton is grazed by zooplankton in the water column. The rate is depending on the phytoplankton (Phyto) and zooplankton (Zoo) biomass and is influenced by the water temperature. The grazing rate is calculated from the maximum specific grazing rate ($`ZooGrazing_{max}`$) [d<sup>-1</sup>] at ambient temperature using a Holling type-II functional response for the phytoplankton.
```math
ZooGrazing = ZooGrazing_{max} \cdot T_{fac} \cdot Zoo \cdot (\frac{Phyto}{Phyto+ks_{ZooGrazing}})
```
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## ZooFaeces
A fraction ($`p_{FaecesZoo}`$) of the uptake of phytoplankton grazed by zooplankton is excreted as faeces into the detritus pool. The rest is used for assimilation.
```math
ZooFaeces = p_{FaecesZoo} \cdot ZooGrazing
```
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## ZooExcretion
Respiration of the zooplankton is described as a first-order response, with $`Resp_{Zoo}`$ as the first order rate constant, and is corrected for water temperature. The nitrogen is excreted in the water column as DIN.
```math
ZooExcretion = Resp_{Zoo} \cdot Zoo \cdot T_{fac}
```
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## ZooMortality
The mortality of the zooplankton is described as a second-order function on the zooplankton biomass. Mortality is also assumed to be temperature-dependent.
```math
ZooMortality = M_{Zoo} \cdot Zoo^{2} \cdot T_{fac}
```
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## MortMPB
In the model, the microphytobenthos is not grazed by filterfeeders. The mortality is described as a second-order function on the microphytobenthos biomass. Mortality is also assumed to be temperature-dependent.
```math
Mort_{MPB} = M_{MPB} \cdot MPB^{2} \cdot T_{fac}
```
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## MortShellfish
The mortality rate ($`Mort_{i}`$, mmol N m-2 d<sup>-1</sup>) of shellfish species i (MUS, OYS and COC) assumed to be density dependent, with $`M_{i}`$ is the specific mortality rate when the biomass of the shellfish species equals the carrying capacity ($`CC_{i}`$). $`DensM_{i}`$ indicates the importance of the density dependent mortality.
```math
Mort_{i} = M_{i} + DensM_{i} \cdot (\frac {FF_{i}}{CC_{i}}-1)) \cdot FF_{i}
```
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## Mineralization
The mineralisation of detritus is described as a first-order function with a first-order rate constant $`r_{Det}`$ The mineralisation rate increases with water temperature.
```math
Mineralization = (r_{Det} \cdot Det \cdot T_{fac}
```
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## DINupMPB
DINupMPB is the production of microphytobenthos [mmolN m<sup>-2</sup> d<sup>-1</sup>]. It is assumed that microphytobenthos only occurs at the intertidal areas. DINupMPB is depending on the irradiation, water temperature, the nutrient concentration (DIN) and the microphytobenthos concentration.
Mean daily irradiation at the water surface [W d<sup>-1</sup>] is derived from the file KNMI measurements (stored in “./DataOS/KNMI/”).
It is assumed that about 50% of the light is photosynthetic active radiation (PAR)
```math
PAR_{b} = 0.5 \cdot I_{0}
```
DINupMPB is calculated from the maximum specific uptake rate [d<sup>-1</sup>] at ambient temperature and the algal concentration. Limitations by light and nutrients are described by Monod formulations.
```math
DIN_{upt,MPB}= MPB_{upt,max} \cdot MPB \cdot (1-\frac{MPB}{CC_{MPB} \cdot f_{int}}) \cdot (\frac{PAR_{b}}{PAR_{b} + ks_{PAR,MPB}}) \cdot (\frac{DIN}{DIN _\cdot + ks_{DIN,MPB}}) \cdot T_{fac}
```
Where $`MPB_{upt,max} `$ is the maximum DIN uptake rate [d<sup>-1</sup>] of microphytobenthos at 10&deg;C, $`ks_{PAR,MPB}`$ and $`ks_{DIN,MPB}`$ are the half-saturation coefficients for light and DIN, respectively. Production of the microphytobenthos is limited by the carrying capacity ($`CC_{MPB}`$, mmol N m<sup>-2</sup>) at the intertidal flat. $`T_{fac}`$ is the correction for the ambient water temperature.
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## BotMin
The mineralisation of bottom detritus is described as a first-order function with a first-order rate constant $`r_{Bot\_Det}`$. The mineralisation rate increases with water temperature.
```math
Mineralization = r_{Bot\_Det} \cdot Bot\_Det \cdot T_{fac}
```
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# Shellfish scope for growth
Processes related to the physiology of the shellfish are described by a scope for growth model.
## Clearance
For each shellfish species (i) the individual clearance rate (〖CR〗_i, l h-1) is calculated using the allometric relation:
〖CR〗_i=a_i∙〖W_i〗^(b_i )
Where W_i is the average weight of the shellfish species i (g AFDW), and a_i and b_i are species-specific coefficients. The total clearance rate (〖Clearance〗_i) [m3 m-2 d-1] for species i is calculated by
〖Clearance〗_i=〖CR〗_i∙N_i∙24/1000
With N_i is the number of individuals per m2 of species i. The factor 24/1000 is to convert from l h-1 to m3 d-1.
The total clearance rate is the sum of the clearance rates of all shellfish species.
Clearance=〖Clearance〗_MUS+〖Clearance〗_OYS+〖Clearance〗_COC
## GrazingPhy
The grazing rate of the shellfish on phytoplankton (〖Grazing〗_(Phy,i)) [mmol N m-2 d-1] is calculated from the clearance rate (〖Clearance〗_i) multiplied by the phytoplankton biomass.
〖Grazing〗_(Phy,i)=〖Clearance〗_i∙Phyto
## GrazingZoo
The grazing rate of the shellfish on zooplankton (〖Grazing〗_Zoo) [mmol N m-2 d-1] is calculated from the clearance rate (Clearance) multiplied by the zooplankton biomass.
〖Grazing〗_(Zoo,i)=〖Clearance〗_i∙Zoo
## GrazingDet
The grazing rate of the shellfish on detritus (〖Grazing〗_Det) [mmol N m-2 d-1] is calculated from the clearance rate (Clearance) multiplied by the detritus biomass.
〖Grazing〗_(Det,i)=〖Clearance〗_i∙Det
## Ingestion
The total concentration of food for the shellfish [mgC m-3] is the sum of phytoplankton, zooplankton and detritus in units of carbon. This is calculated from Phyto, Zoo and Det, respectively, from the molecular weight of C and assuming a fixed C/N ratio. The total food concentrations in units of C is calculated from:
FoodC=PhyC+ZooC+DetC
Ingestion rate [mg C d-1] for each inidivdual of species i is a function of the size of the shellfish, the food concentration described by a Holling type-II functional response.
〖Ingestion〗_i=I_(m,i)∙〖W_i〗^(b_(m,i) )∙FoodCk_i+FoodC∙Tfac
Where I_(m,i) is the reference ingestion rate for shellfish i, W_i is the mean individual weight [mg AFDW] and b_(m,i) is the allometric exponent for ingestion. k_i is the half-saturation coefficient for shellfish i [mg C m-3],
## Assimilation
A fraction of the ingestion is used for assimilation [mg C d-1]:
〖Assimilation〗_i=ε_i∙〖Ingestion〗_i
With ε_i is the dimensionless species specific feeding efficiency.
ε_i=((ε_(i,PhyZoo)∙(PhyC+ZooC)+ε_(i,Det)∙μ_i∙DetC)/((PhyC+ZooC)+μ_i∙DetC))
ε_(i,PhyZoo) is the dimensionless phytoplankton and zooplankton assimilation efficiency for shellfish species i and μ_i is the dimensionless feeding preference coefficient for species i. PhyC, ZooC and DetC are phytoplankton, zooplankton and detritus concentration in units of gC m-3.
## Respiration
Respiration rate is the CO2 production by an individual organism [mg C d-1] and is composed of basal respiration (R_(B,i)) and costs for growth (R_(G,i)).
〖Respiration〗_i=R_(B,i)+ R_(G,i)
The costs for growth is assumed to be a fixed fraction of the assimilation rate:
R_(G,i)=σ_i∙〖Assimilation〗_i
Basal respiration is a function of the size of the animal and increased with the water temperature:
R_(B,i)=β_(RS,i)∙〖W_i〗^(b_(m,i) )∙T_fac
β_(RS,i) is the standard respiration rate for species i [d-1] and b_(m,i) is a coefficient.
Respiration rate is expressed in units of mg C d-1.
## DINprod
〖DINprod〗_i is the respiration rate of the total shellfish stock of a particular species within the compartment in units of mmol N m-2 d-1 and is calculated from the respiration rate of an individual organism.
〖DINprod〗_i=(〖Respiration〗_i∙N_i)/(C⁄N∙M_c )
Where N_i is the number of shellfish species i per m2, C⁄N is the C/N ratio of the food and M_c is the molar weight for carbon (g mole-1).
## Growth
Growth [mmol N m-2 d-1] of the total stock is calculated from the difference between the individual assimilation rate [mg C d-1] and respiration rate [mg C d-1]. The difference is multiplied by the average density of the species within the compartment (N_i) [# m-2] and corrected for the C/N ratio and the molar weight of C to obtain growth in units of mmol N m-2 d-1.
〖Growth〗_i=(〖Assimilation〗_i-〖Respiration〗_i )∙(N_i/(C⁄N∙M_c ))
## Faeces
Faeces production is of the total stock [mmol N m-2 d-1] is calculated from the difference between individual ingestion rate [mg C d-1] and assimilation rate [mg C d-1]. The difference is multiplied by the average density of the species within the compartment (N_i) [# m-2] and corrected for the C/N ratio and the molar weight of C to obtain faeces production in units of mmol N m-2 d-1.
〖Faeces〗_i=(〖Ingestion〗_i-〖Assimilation〗_i )∙(N_i/(C⁄N∙M_c ))
## Pseudofaeces
Pseudofaeces production of the total stock is calculated from the difference between individual filtration rate [mg C d-1] and ingestion rate [mg C d-1]. The difference is multiplied by the average density of the species within the compartment (N_i) [# m-2] and corrected for the C/N ratio and the molar weight of C to obtain pseudofaeces production in units of mmol N m-2 d-1.
〖Pseudo_faeces〗_i=(〖Filtration〗_i-〖Ingestion〗_i )∙(N_i/(C⁄N∙M_c ))
Where individual filtration rate [mg C d-1] is calculated from the total clearance rate [m3 m-2 d-1] and the amount of food [mgC m-3]
〖Filtration〗_i=(〖Clearance〗_i∙FoodC)/N_i
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