Biogeochemical processes

Soil temperature

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).

Nitrogen mass balance

\[\frac{\partial NO_{3-soil}(T,t)}{\partial t} + \frac{\partial NO_{3-soil}(T,t)}{\partial T} = 0.3 \cdot N_{fert}(t) + Nit_{soil}(t) - Denit_{soil}(t) - N_{up}(t)\]

\[\frac{\partial Nmin(T,t)}{\partial t} + \frac{\partial Nmin(T,t)}{\partial T} = 0.7 \cdot N_{fert}(t) + Min_{soil}(t) - Nit_{soil}(t)\]

Denitrification

Denitrification rate of the soil \(Denit_{soil}\) (\(kg N ha^{-1} year^{-1}\)) at time step \(t\):

\[\begin{split}Denit_{soil}=\begin{cases} (Denit_{soil-max} \cdot \frac{NO_{3-soil}(T,t)}{k_{denit-soil} + NO_{3-soil}(T,t)}) \cdot c_{denit} & S_{t} \geq 0.7 \cdot S_{sat}\\ 0 & S_{t} < 0.7 \cdot S_{sat} \end{cases}\end{split}\]

with

\[\begin{split}c_{denit}=\begin{cases} \frac{TA_{soil}}{50 - 5} & 5 \leq TA_{soil} \leq 50 \\ 0 & 5 > TA_{soil} \lor TA_{soil} > 50 \end{cases}\end{split}\]

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

Nitrification rate of the soil \(Denit_{soil}\) (\(kg N ha^{-1} year^{-1}\)) at time step \(t\):

\[\begin{split}Nit_{soil}=\begin{cases} Nit_{soil}=(Nit_{soil-max} \cdot \frac{Nmin(T,t)}{k_{nit} + Nmin(T,t)}) \cdot c_{nit} & S_{t} < 0.7 \cdot S_{sat}\\ 0 & S_{t} > 0.7 \cdot S_{sat} \end{cases}\end{split}\]

with

\[\begin{split}c_{nit}=\begin{cases} \frac{TA_{soil}}{50 - 5} & 5 \leq TA_{soil} \leq 50 \\ 0 & 5 > TA_{soil} \lor TA_{soil} > 50 \end{cases}\end{split}\]

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

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

\[\begin{split}c_{min}=\begin{cases} \frac{TA_{soil}}{50 - 5} & 5 \leq TA_{soil} \leq 50 \\ 0 & 5 > TA_{soil} \lor TA_{soil} > 50 \end{cases}\end{split}\]

where \(k_{min}\) is the maximum nitrogen mineralization rate of the soil (\(kg N ha^{-1} year^{-1}\)),

Nitrogen uptake by crops

Nitrogen uptake by crops \(N_{up}\) (mg \(\Delta t^{-1}\)) at time step \(t\):

\[N_{up}= transp \cdot \int_{T=0}^{\infty} C_S(T,t) \cdot \alpha_p \cdot \overleftarrow{p}_{transp}(T,t) dT\]