Soil temperature TA_{soil} (°C) at time step t:
TA_{soil}=TA_{year} + (TA - TA_{year}) \cdot \frac{sin(\omega t + \omega \Phi -\frac{0.5 \cdot z_{soil}}{D})}{e^\frac{-0.5 \cdot z_{soil}}{D}}
with
\omega = \frac{2\pi}{365}
where TA_{year} is the annual average air temperature (°C), TA is the daily air temperature (°C) at time step t, D is the dampening depth (mm), t is the day of year and \Phi is a constant offset (days).
\frac{\partial NO_{3-soil}(T,t)}{\partial t} + \frac{\partial NO_{3-soil}(T,t)}{\partial T} = 0.3 \cdot N_{fert}(t) + Nit_{soil}(t) - Denit_{soil}(t) - N_{up}(t)
\frac{\partial Nmin(T,t)}{\partial t} + \frac{\partial Nmin(T,t)}{\partial T} = 0.7 \cdot N_{fert}(t) + Min_{soil}(t) - Nit_{soil}(t)
Denitrification rate of the soil Denit_{soil} (kg N ha^{-1} year^{-1}) at time step t:
Denit_{soil}=\begin{cases}
(Denit_{soil-max} \cdot \frac{NO_{3-soil}(T,t)}{k_{denit-soil} + NO_{3-soil}(T,t)}) \cdot c_{denit} & S_{t} \geq 0.7 \cdot S_{sat}\\
0 & S_{t} < 0.7 \cdot S_{sat}
\end{cases}
with
c_{denit}=\begin{cases}
\frac{TA_{soil}}{50 - 5} & 5 \leq TA_{soil} \leq 50 \\
0 & 5 > TA_{soil} \lor TA_{soil} > 50
\end{cases}
where Denit_{soil-max} is the maximum dentrification rate (kg N ha^{-1} year^{-1}), k_{denit-soil} is the Michaelis constant (kg N ha^{-1} year^{-1}), NO_{3-soil} is the age-ranked nitrate nitrogen storage of the soil (kg NO_{3}-N), S is the storage volume (mm) at time step t and S_{sat} is the storage volume at saturation (mm).
Denitrification rate of the groundwater Denit_{gw} (kg N ha^{-1} year^{-1}) at time step t:
Denit_{gw}=Denit_{gw-max} \cdot \frac{NO_{3-gw}(T,t)}{k_{denit-gw} + NO_{3-gw}(T,t)}
where Denit_{gw-max} is the maximum dentrification rate (kg N ha^{-1} year^{-1}), k_{denit-gw} is the Michaelis constant (kg N ha^{-1} year^{-1}), NO_{3-gw} is the age-ranked nitrate nitrogen storage of the groundwater (kg NO_{3}-N),
Nitrification rate of the soil Denit_{soil} (kg N ha^{-1} year^{-1}) at time step t:
Nit_{soil}=\begin{cases}
Nit_{soil}=(Nit_{soil-max} \cdot \frac{Nmin(T,t)}{k_{nit} + Nmin(T,t)}) \cdot c_{nit} & S_{t} < 0.7 \cdot S_{sat}\\
0 & S_{t} > 0.7 \cdot S_{sat}
\end{cases}
with
c_{nit}=\begin{cases}
\frac{TA_{soil}}{50 - 5} & 5 \leq TA_{soil} \leq 50 \\
0 & 5 > TA_{soil} \lor TA_{soil} > 50
\end{cases}
where Nit_{soil-max} is the maximum nitrification rate (kg N ha^{-1} year^{-1}), k_{nit-soil} is the Michaelis constant (kg N ha^{-1} year^{-1}) and Nmin is the age-ranked mineral nitrogen storage of the soil (kg N).
Nitrogen mineralization rate of the soil Min_{soil} (kg N ha^{-1} year^{-1}) at time step t:
Min_{soil}=k_{min} \cdot c_{min}
with
c_{min}=\begin{cases}
\frac{TA_{soil}}{50 - 5} & 5 \leq TA_{soil} \leq 50 \\
0 & 5 > TA_{soil} \lor TA_{soil} > 50
\end{cases}
where k_{min} is the maximum nitrogen mineralization rate of the soil (kg N ha^{-1} year^{-1}),
Nitrogen uptake by crops N_{up} (mg \Delta t^{-1}) at time step t:
N_{up}= transp \cdot \int_{T=0}^{\infty} C_S(T,t) \cdot \alpha_p \cdot \overleftarrow{p}_{transp}(T,t) dT