# HYDRUS-1D Simulation of Nitrogen Dynamics in Rainfed Sweet Corn Production
## 2 Material and Methods
### HYDRUS-ID Model  
#### 2.2.1 Model Description  
The one-dimensional numerical model HYDRUS-1D [11] was used to simulate soil water and solute dynamics in the sweet corn field. The model is capable of simultaneously considering the transport of multiple solutes that are subject to first-order degradation. The governing transport equations were solved using a Galerkin-type linear finite element scheme.  
The partial differential equations for one-dimensional advective-dispersive N transport for a first-order chain reaction in variably saturated media is given as [12].  
Total Nitrogen (TN):  
\[\frac{\partial\partial C_{1}}{\partial t}=\frac{\partial}{\partial z}\bigg{(} \partial D_{1}^{w}\frac{\partial C_{1}}{\partial z}\bigg{)}-\frac{\partial qC _{1}}{\partial z}-\mu_{w,1}^{\prime}\theta C_{1} \tag{2}\]  
Ammonium nitrate (NH4\({}^{+}\)-N):  
\[\begin{split}\frac{\partial\theta C_{2}}{\partial t}+\frac{ \partial\rho S_{2}}{\partial t}+\frac{\partial a_{v}g_{2}}{\partial t}=\frac{ \partial}{\partial z}\bigg{(}\partial D_{2}^{w}\frac{\partial C_{2}}{ \partial z}\bigg{)}+\frac{\partial}{\partial z}\bigg{(}a_{v}D_{2}^{g}\frac{ \partial g_{2}}{\partial z}\bigg{)}-\frac{\partial qC_{2}}{\partial z}\\ -\mu_{w,2}^{\prime}\theta C_{2}+\mu_{w,1}^{\prime}\theta C_{1}+ \gamma_{s,2}\rho-r_{a,2}\end{split} \tag{3}\]  
Nitrate (NO3\({}^{-}\)-N):  
\[\frac{\partial\theta C_{3}}{\partial t}=\frac{\partial}{\partial z}\bigg{(} \partial D_{3}^{w}\frac{\partial C_{3}}{\partial z}\bigg{)}-\frac{\partial qC _{3}}{\partial z}-\mu_{w,3}^{\prime}\theta C_{3}+\mu_{w,2}^{\prime}\theta C_{ 2}-r_{a,3} \tag{4}\]  
where, \(C\) is the solute concentration in liquid phase (mg\(\cdot\)L\({}^{-1}\)), \(S\) is the solute concentration in the solid phase (mg\(\cdot\)g\({}^{-1}\)), \(g\) is the solute concentration in gas phase (mg\(\cdot\)g\({}^{-1}\)), \(\rho\) is the dry bulk density (g\(\cdot\)cm\({}^{-3}\)), \(\theta\) is the volumetric water content (cm\({}^{3}\)cm\({}^{-3}\)), \(q\) is the volumetric flux density (cm\(\cdot\)day\({}^{-1}\)), \(\mu_{w}\) is the first-order rate constant for solute in the liquid phase (day\({}^{-1}\)), \(\mu_{w}\) is the first-order rate constant for chain reaction (day\({}^{-1}\)), \(\gamma_{s}\) is a zero-order rate constant in the solid phase (day\({}^{-1}\)), \(D_{w}\) is the dispersion coefficient for liquid phase (cm\({}^{2}\)\(\cdot\)day\({}^{-1}\)), \(D_{g}\) is the dispersion coefficient for gas phase (cm\({}^{2}\)\(\cdot\)day\({}^{-1}\)), \(r_{a}\) is the root nutrient uptake. The subscripts 1,2 and 3 represents N species urea, NH4\({}^{+}\)-N and NO3\({}^{-}\)-N, respectively. A linear Equation (5) describes the adsorption isotherm relating S\({}_{2}\) and C\({}_{2}\).  
\[S_{2}=k_{d,2}C_{2} \tag{5}\]  
where \(k_{d,2}\) is the distribution coefficient for NH4\({}^{+}\)-N (L\(\cdot\)mg\({}^{-1}\)).  
The N transformation processes considered here include; hydrolysis of N species to NH4\({}^{+}\)-N, nitrification of NH4\({}^{+}\)-N to NO3\({}^{-}\)-N, denitrification of NO3\({}^{-}\)-N to N2/N2O, and root uptake of NH4\({}^{+}\)-N and NO3\({}^{-}\)N, where; hydrolysis, nitrification, and denitrification were considered as first-order reactions [27].The denitrification process was ignored in the study. The details of the simulation process and model description are presented in the flow chart (Figure 5).