# Nutrient Phytoplankton Zooplankton Detritus (NPZD) model

The provided NPZD model is the low complexity model of Kuhn et al. (2015).

## Model equations

$$$\frac{\partial P}{\partial t} = \mu_\text{max}\frac{N}{k_N + N}\frac{\alpha PAR}{\sqrt{\mu_\text{max}^2 + \alpha^2PAR^2}}P - g_\text{max}\frac{P^2}{k_P^2 + P^2}Z-(l_{PN}+l_{PD})P,$$$$$$\frac{\partial Z}{\partial t} = \beta g_\text{max}\frac{P^2}{k_P^2 + P^2}Z - (l_{ZN} + l_{ZD}Z)Z,$$$$$$\frac{\partial D}{\partial t} = (1 - \beta) g_\text{max}\frac{P^2}{k_P^2 + P^2}Z + l_{ZD}Z^2 + l_{PD}P - r_{DN}D,$$$$$$\frac{\partial N}{\partial t} = - \mu_\text{max}\frac{N}{k_N + N}\frac{\alpha PAR}{\sqrt{\mu_\text{max}^2 + \alpha^2PAR^2}}P + l_{PN}P + l_{ZN}Z + r_{DN}D.$$$

Here $\mu_\text{max} = \mu_0Q_{10}(T)$ and $l_{XY} = l_{XY0} Q_{10}(T)$ where $Q_{10}(T) = 1.88^{T/10}$.

Additionally, the phytoplankton and detritus sink at a constant rate.

### Parameter variable names

SymbolVariable nameUnits
$\alpha$initial_photosynthetic_slope1 / (W / m² / s)
$\mu_0$base_maximum_growth1 / s
$k_N$nutrient_half_saturationmmol N / m³
$l_{PN0}$base_respiration_rate1 / s
$l_{PD0}$phyto_base_mortality_rate1 / s
$g_\text{max}$maximum_grazing_rate1 / s
$k_P$grazing_half_saturationmmol N / m³
$\beta$assimulation_efficiency-
$l_{ZN}$base_excretion_rate1 / s
$l_{ZD}$zoo_base_mortality_rate1 / s
$r_{DN}$remineralization_rate1 / s

All default parameter values are given in Parameters.

## Model conservation

Nitrogen is conserved in the evolution of this model (excluding external sources and sinking), i.e. $\frac{\partial P}{\partial t} + \frac{\partial Z}{\partial t} + \frac{\partial D}{\partial t} + \frac{\partial N}{\partial t} = 0$.