Modeling nitrate leaching on a cropped Andosol

Abstract

The nitrate transfer model in Andosol using the multipurpose solver was developed using meteorological characteristics, three-species nitrification and/or denitrification chain (NH₄ → NO₃ → N₂), and a cropping system to evaluate the behavior of nitrate leaching from a cropped Andosol. Its validity was then tested by comparing results calculated by proposed model with results from a capillary lysimeter experiment. The measured values of cumulative water flux and cumulative nitrate nitrogen (NO₃-N) flux were compared with those of simulated values, and the accuracy of proposed model was verified by using test statistics of the root mean square error (RMSE) and t-value. The calculated results using the proposed model were accurate despite the small numbers of parameters (e.g. denitrification rate, immobilization rate etc.) and simple mechanistic sub-models were adopted. To understand the behavior of NO₃-N concentration visually and qualitatively in the soil horizon, we used colored isopleths of NO₃-N concentration in pore water at a cropped Andosol. From the validated results and isopleths results, the leaching risk of nitrate is likely to increase when (1) the agricultural land has high permeability and low capability of adsorption of nitrate, and when (2) heavy rains fall soon after application of nitrogen fertilizer. The results of RMSE and t-test, it was demonstrated that relationship between FlexPDE and observed data is in more agreement than the Hydrus-1D model.

Appendices

Appendices

Soil sub-model

Soil water retention function and unsaturated hydraulic conductivity are expressed as Eqs. A1 and A2, respectively (van Genuchten 1980);

$$ \theta (\psi ) = \theta_{\text{r}} + \frac{{\theta_{\text{s}} - \theta_{\text{r}} }}{{[1 + (\alpha |\psi |)^{n} ]^{m} }},\;m = 1 - \frac{1}{n} $$

(A1)

$$ K_{\text{u}} (\theta ) = K_{\text{s}} \Uptheta^{1/2} [1 - (1 - \Uptheta^{1/m} )^{m} ]^{2} ,\,\,\,\,\Uptheta = \frac{{\theta - \theta_{\text{r}} }}{{\theta_{\text{s}} - \theta_{\text{r}} }} $$

(A2)

where, θ: volumetric water content (cm³ cm⁻³), θ_s: saturated volumetric water content (cm³ cm⁻³), θ_r: residual volumetric water content (cm³ cm⁻³), ψ: matric potential (kPa), α: shape parameter (kPa⁻¹), n: scaling parameter, K _u: unsaturated hydraulic conductivity (cm s⁻¹) and Θ: effective saturation degree.

The Langmuir type adsorption isotherm was adopted to describe the relationship between solution concentration of pore water and adsorbed solute (Langmuir 1918). Ammonium nitrogen and nitrate nitrogen are expressed as Eqs. A3 and A4 as follows:

$$ s_{1} (c_{1} ) = (k_{1}^{{{\text{NH}}_{4} {\text{ - N}}}} c_{1} )/(1 + k_{2}^{{{\text{NH}}_{4} {\text{ - N}}}} c_{1} ) $$

(A3)

$$ s_{2} (c_{2} ) = (k_{1}^{{{\text{NO}}_{3} {\text{ - N}}}} c_{2} )/(1 + k_{2}^{{{\text{NO}}_{3} {\text{ - N}}}} c_{2} ) $$

(A4)

where, s ₁: adsorbed ammonium nitrogen (mg g⁻¹), c ₁: concentration of ammonium nitrogen (mg cm⁻³), s ₂: adsorbed nitrate nitrogen (mg g⁻¹), c ₂: concentration of nitrate nitrogen (mg cm⁻³), k ₁ and k ₂ is fitting parameters of each of nitrogen. The maximum adsorption s _max can be expressed as ratio of k ₁ and k ₂ (s _max = k ₁/k ₂).

The small amount of adsorption of nitrate nitrogen at z = 10 cm and z = 30 cm were observed under lower solute concentration condition. Thus, we adopted sigmoid type equation instead of (A4) expressed as follows;

$$ s_{2} (c_{2} ) = (\delta c_{2}^{0.2} )/[1 + { \exp }( - \varepsilon (c_{2} - \eta ))] $$

(A5)

where, δ, ε and η are empirical parameters listed in Table 2.

The DPHP technique for determining soil thermal properties is based on an analytical solution for the radial heat conduction equation with an infinite line heat source in an isotropic and homogeneous medium at a uniform initial temperature. Temperature change with time after a heat pulse emitted can be described as (Kluitenberg et al. 1993);

$$ \Updelta T(r,t) = \frac{Q}{4\pi \kappa }\left[ {Ei\left( {\frac{{ - r^{2} }}{{4\kappa \left( {t - t_{0} } \right)}}} \right) - Ei\left( {\frac{{ - r^{2} }}{4\kappa \cdot t}} \right)} \right]\quad (t > t_{0} ) $$

(A6)

where, ΔT: temperature change (K), t: time (s), t ₀: heating period (s), r: radial distance (m), κ: thermal diffusivity (m² s⁻¹), −Ei(−x): the exponential integral and Q: source strength per unit length per unit time (K m² s⁻¹), Q is equivalent to q/ρc where q: quantity of heat librated per unit length (W m⁻¹), ρc: volumetric heat capacity (MJ m⁻³ K⁻¹).

Bristow et al. (1994) led to following solutions for ρc and κ.

$$ \rho c = \frac{q}{{4\pi \kappa \Updelta T_{\text{m}} }}\left[ {Ei\left( {\frac{{ - r^{2} }}{{4\kappa \left( {t_{\text{m}} - t_{0} } \right)}}} \right) - Ei\left( {\frac{{ - r^{2} }}{{4\kappa t_{\text{m}} }}} \right)} \right] $$

(A7)

$$ \kappa = \frac{{r^{2} }}{4}\left[ {\frac{{(1/(t_{\text{m}} - t_{0} )) - (1/t_{\text{m}} )}}{{\ln \,(t_{\text{m}} /(t_{\text{m}} - t_{0} ))}}} \right] $$

(A8)

where, ΔT _m: maximum temperature change (K), t _m: time (s) when ΔT _m was observed. Thermal conductivity λ was calculated by ρc × κ.

In order to parameterize the thermal properties of the soil, the model of de Vries (1963) was used for volumetric heat capacity which expressed as a function of volumetric water content. Thermal conductivity was also fitted with first order function of volumetric water content as follows:

$$ \rho c = \rho c_{\text{w}} \theta + \rho c_{0} $$

(A9)

$$ \lambda = \lambda_{\text{w}} \theta + \lambda_{0} $$

(A10)

where, ρc _w: slope between volumetric water content and volumetric heat capacity (MJ m⁻³ K⁻¹), ρc ₀: volumetric heat capacity under oven dried condition (MJ m⁻³ K⁻¹), λ: thermal conductivity of bulk soil (W m⁻¹ K⁻¹), λ_w: slope between volumetric water content and thermal conductivity (W m⁻¹ K⁻¹) and λ₀: intercept of regressed line (W m⁻¹ K⁻¹).

Crop sub-model

The relative amount of N-uptake for crop growing expressed as a function of time was calculated by fitting the observed data (Ibaraki Agricultural Center 1991; Fukuoka Agricultural Research Center 2003; Chiba Prefectural 2005; Suzuki 1996) to Eq. A11,

$$ U(t) = \left\{ {\begin{array}{*{20}c} {\sum\limits_{i = 0}^{n} {a_{i} (t - t_{s} )^{i} \quad (i = 0,1, \ldots ,n)} } \hfill & {{\text{for}}\;{\text{carrots}},\;{\text{spinach}}\;{\text{and}}\;{\text{sweet}}\;{\text{corn}}} \hfill \\ {1/1 + \exp \,[ - K(t - t_{s} - m)]} \hfill & {\text{otherwise}} \hfill \\ \end{array} } \right. $$

(A11)

where, U(t): relative amount of N-uptake, i: order of polynomial equation, a: coefficient of i-th degree polynomial equation, t: elapsed day (days), t _s: beginning of inorganic nitrogen uptake (days), K and m: fitting parameters.

The crop root grows downward with days. To express the nitrogen uptake from the crop root, we assumed the rooting depth function R(t) (cm) in Eq. A12,

$$ R(t) = R_{ \max } \,U(t)\quad (5 \le R < R_{ \max } ) $$

(A12)

where, R _max is the maximum rooting depth (cm).

Water and nutrient uptake distribution S(R, z) was calculated using Eq. A13,

$$ S(R,z) = \left\{ {\begin{array}{*{20}l} {2.05\left\{ {\frac{1}{{2R_{ \max } }} - [R(1 + { \exp }(z))]^{ - 1} } \right\}} & {(z < R)} \\ 0 & {\text{otherwise}} \\ \end{array} } \right. $$

(A13)

where, z is distance from soil surface (cm).

NH₄-N and NO₃-N concentrations of pore water within rooting depth are expressed in Eqs. A14 and A15, respectively,

$$ c_{{{\text{NH}}_{ 4} {\text{ - N}}}}^{\text{AVG}} = \frac{1}{R(t)}\int_{z = 0}^{z = R(t)} {c_{{{\text{NH}}_{ 4} {\text{ - N}}}} (z){\text{d}}z} $$

(A14)

$$ c_{{{\text{NO}}_{3} {\text{ - N}}}}^{\text{AVG}} = \frac{1}{R(t)}\int_{z = 0}^{z = R(t)} {c_{{{\text{NO}}_{3} {\text{ - N}}}} (z){\text{d}}z} $$

(A15)

where c(z) is the inorganic nitrogen concentration of pore water expressed as a function of z (mg cm⁻³) and c ^AVG is the average inorganic nitrogen concentration (mg cm⁻³) within R(t).

From Eqs. A14 and A15, the control functions of NH₄-N and NO₃-N uptake were derived as Eqs. A16 and A17, respectively,

$$ u_{{{\text{NH}}_{4} {\text{ - N}}}} (z) = \left\{ {\begin{array}{*{20}c} {\frac{{N_{\text{uptake}} c_{{{\text{NH}}_{4} {\text{ - N}}}}^{\text{AVG}} }}{{c_{{{\text{NH}}_{4} {\text{ - N}}}}^{\text{AVG}} + c_{{{\text{NO}}_{3} {\text{ - N}}}}^{\text{AVG}} }}} \hfill & {\left( {c_{{{\text{NH}}_{4} {\text{ - N}}}} (z) \ge c_{{{\text{NH}}_{4} {\text{ - N}}}}^{ \max } } \right)} \hfill \\ {\left( {\frac{{c_{{{\text{NH}}_{4} {\text{ - N}}}} (z) - c_{{{\text{NH}}_{4} {\text{ - N}}}}^{ \min } }}{{c_{{{\text{NH}}_{4} {\text{ - N}}}}^{ \max } - c_{{{\text{NH}}_{4} {\text{ - N}}}}^{ \min } }}} \right)\frac{{N_{\text{uptake}} c_{{{\text{NH}}_{4} {\text{ - N}}}}^{\text{AVG}} }}{{c_{{{\text{NH}}_{4} {\text{ - N}}}}^{\text{AVG}} + c_{{{\text{NO}}_{3} {\text{ - N}}}}^{\text{AVG}} }}} \hfill & {(c_{{{\text{NH}}_{4} {\text{ - N}}}}^{ \min } < c_{{{\text{NH}}_{4} {\text{ - N}}}} (z) < c_{{{\text{NH}}_{4} {\text{ - N}}}}^{ \max } )} \hfill \\ 0 \hfill & {\text{otherwise}} \hfill \\ \end{array} } \right. $$

(A16)

$$ u_{{{\text{NO}}_{3} {\text{ - N}}}} (z) = \left\{ {\begin{array}{*{20}c} {\frac{{N_{\text{uptake}} c_{{{\text{NO}}_{3} {\text{ - N}}}}^{\text{AVG}} }}{{c_{{{\text{NH}}_{4} {\text{ - N}}}}^{\text{AVG}} + c_{{{\text{NO}}_{3} {\text{ - N}}}}^{\text{AVG}} }}} \hfill & {(c_{{{\text{NO}}_{3} {\text{ - N}}}} (z) \ge c_{{{\text{NO}}_{3} {\text{ - N}}}}^{ \max } )} \hfill \\ {\left( {\frac{{c_{{{\text{NO}}_{3} {\text{ - N}}}} (z) - c_{{{\text{NO}}_{3} {\text{ - N}}}}^{\min } }}{{c_{{{\text{NO}}_{3} {\text{ - N}}}}^{ \max } - c_{{{\text{NO}}_{3} {\text{ - N}}}}^{\min } }}} \right)\frac{{N_{\text{uptake}} c_{{{\text{NO}}_{3} {\text{ - N}}}}^{\text{AVG}} }}{{c_{{{\text{NH}}_{4} {\text{ - N}}}}^{\text{AVG}} + c_{{{\text{NO}}_{3} {\text{ - N}}}}^{\text{AVG}} }}} \hfill & {(c_{{{\text{NO}}_{3} {\text{ - N}}}}^{ \min } < c_{{{\text{NO}}_{3} {\text{ - N}}}} (z) < c_{{{\text{NO}}_{3} {\text{ - N}}}}^{ \max } )} \hfill \\ 0 \hfill & {\text{otherwise}} \hfill \\ \end{array} } \right. $$

(A17)

where u(z) is nitrogen uptake amount at the arbitrary position z (mg cm⁻²), N _uptake is the set value of nitrogen uptake by crop roots (mg cm⁻²), c ^max is the upper limit concentration threshold to uptake (=5.0 × 10⁻⁴ mg cm⁻³) and c ^min is the lower limit concentration threshold to uptake (≓0 mg cm⁻³). The advantage of using Eqs. A16 and A17 is that the concentration of pore water does not become a negative value unless the FEM solution is not stable (shaking of FEM solution).

Mineralization, nitrification and denitrification sub-model

Nitrification rate k _nit (derive from fertilizer application) can be expressed in Eqs. A18 and A19,

$$ k_{\text{nit}} (t,T) = 1.1c(T){ \exp }( - ct) $$

(A18)

$$ c(T) = 4.757 \times 10^{ - 3} { \exp }(0.125T) $$

(A19)

where, k _nit: nitrification rate expressed as function of time and temperature (day⁻¹), t: time (days), T: soil temperature (°C). Coefficient of right hand side of Eqs. A18 and A19 are based on Itahashi et al. (2006).

Nitrification rate $k^{\prime}_{\text{nit}}$ (derive from mineralized nitrogen) is expressed as first derivative of DTS function with respect to time, and is given in Eq. A20,

$$ k^{\prime}_{\text{nit}} = \frac{1.027}{{1 + { \exp }[ - K(t - m)]^{2} }}{\text{exp[}} - K(t - m) ] $$

(A20)

where, K: fitting parameter (=0.0224), m: fitting parameter (198.54 (d)).

Mineralization and immobilization are arise from z < 20 (cm), and distribution function γ(z) is expressed in Eq. A21,

$$ \gamma (z) = \frac{{K^{\prime}}}{{1 + \exp [m^{\prime}(z - 1)]}}\quad (0 < z \le 20({\text{cm}}))\quad \because \quad \int_{{z = 0\,{\text{cm}}}}^{{z = 20\,{\text{cm}}}} {\gamma (z){\text{d}}z \approx 1} $$

(A21)

where, K and m is fitting parameters (K′ = 0.09, m′ = 0.1569) based on Fujitomi et al. (2005), as 0.09 and 0.1569, respectively.

Meteorological sub-model

The daily variation of precipitation for calculating period (monthly) is expressed as a Fourier series (Tashiro 1986),

$$ P_{0} (t) = \frac{{a_{0} }}{2} + \sum\limits_{n = 1}^{50} {\left[ {a_{n} { \cos }\left( {\frac{n\pi (t - \zeta )}{\zeta }} \right) + b_{n} { \sin }\left( {\frac{n\pi (t - \zeta )}{\zeta }} \right)} \right]} $$

(A22)

$$ a_{n} = \frac{1}{\zeta }\int_{ - \zeta }^{\zeta } {R_{0} (t){ \cos }\left( {\frac{n\pi t}{\zeta }} \right){\text{d}}t} \quad (n = 0,1, \ldots ,50) $$

(A23)

$$ b_{n} = \frac{1}{\zeta }\int_{ - \zeta }^{\zeta } {R_{0} (t){ \sin }\left( {\frac{n\pi t}{\zeta }} \right){\text{d}}t} \quad (n = 1,2, \ldots ,50) $$

(A24)

where P ₀(t): daily precipitation (cm day⁻¹), t: elapsed days (days), a _n: Fourier’s cosine coefficient, b _n: Fourier’s sine coefficient, and ζ: half of the periodic interval of month (=15 days).

Temperature amplitude at the soil surface was calculated, and surface temperature was expressed as a function of elapsed days using Eqs. A25–A27 (Campbell and Norman 2003);

$$ T_{0} (t) = T_{\text{avg}} (t) + A_{0} { \sin }\left( {2\pi t + \frac{\pi }{2}} \right) $$

(A25)

$$ A_{0} = A_{{10\,{\text{cm}}}} \left[ {{ \exp }\left( {\frac{{z_{{10\,{\text{cm}}}} }}{D}} \right)} \right]^{ - 1} $$

(A26)

$$ D = \sqrt {\frac{2\,\kappa }{\omega }} = \sqrt {\frac{{2\,\lambda \,\rho c^{ - 1} }}{\omega }} $$

(A27)

where T ₀(t): surface temperature (°C), t: elapsed days (days), T _avg(t): average soil temperature at z = 10 cm depth (°C), A ₀: temperature amplitude at the surface, A _10 cm: temperature amplitude at depth of z = 10 cm, z _10 cm: z = 10 cm depth, D: damping depth (m), λ: thermal conductivity (W m⁻¹ K⁻¹), ρc: volumetric heat capacity (MJ m⁻³ K⁻¹) and ω: angular frequency (=2π/86400 (s)).