Matter transport

In soil, nutrients are transported with the moving water. So far, MONICA only describes the translocation of dissolved nitrate ions. However, using the convection-dispersion equation, modelling of ammonium, sulphate, and dissolved organic matter transport is also imaginable. The basic version of the convection-dispersion equation is:

\[\frac{\partial \theta c}{\partial t} = \frac{\partial}{\partial z} \cdot \left ( \theta \cdot D \cdot \frac{\partial c}{\partial z}\right ) - \frac{\partial q c}{\partial z}\]

\(\theta\) Soil moisture content \([m^{3} \, m^{-3}]\)
\(c\) Solute concentration \([kg \, m^{-3}]\)
\(D\) Effective dispersion coefficient \([m^{2} \, m^{-1}]\)
\(z\) Soil depth \([m]\)
\(t\) Time \([d]\)

which includes water flow:

\[q = -K (\Psi_m) \cdot \left ( \frac{\partial \Psi_m}{\partial z} - 1 \right)\]

\(q\) Water flux density \([m \, d^{-1}]\)
\(\Psi_m\) Soil matrix potential \([cm \, WS]\)
\(K(\Psi_m)\) Hydraulic conductivity \([m \, d^{-1}]\)
\(\theta\) Soil moisture content \([m^3 \, m^{-3}]\)
\(z\) Soil depth \([m]\)

the effective dispersion coefficient:

\[D = D_0 \cdot \frac{1}{\tau} + \alpha_v \cdot \left | \frac{q}{\theta} \right |\]

\(D\) Effective dispersion coefficient \([m^2 \, d^{-1}]\)
\(D_0\) Diffusion coefficient in solution (0.000214) \([m^2 \, d^{-1}]\)
\(\tau\) Tortuosity \([m^2 \, m^{-2}]\)
\(\alpha_v\) Dispersion factor (0.05) \([m]\)
\(q\) Water flux density \([m \, d^{-1}]\)
\(\theta\) Volumetric water content \([m^3 \, m^{-3}]\)

in which tortuosity is used as its inverse, pore space continuity.

\[\tau = \frac{\theta}{a \cdot e^{10 \cdot \theta}}\]

\(\tau\) Tortuosity \([m^2 \, m^{-2}]\)
\(\theta\) Volumetric water content \([m^3 \, m^{-3}]\)
\(a\) Factor (0.002)

The convection-dispersion equation is solved using the explicit finite difference method.

For \(q1/2 > 0\):

\[\frac{ \theta_{i , t+\Delta t} \cdot c_{i, t+\Delta t} - \theta_{i, t} \cdot c_{i, t}} {\Delta t} =\]

\[\left ( \theta_{i - \frac{1}{2}, t} \cdot D_{i-\frac{1}{2}, t} - \frac{\Delta z}{2} \cdot q_{i-\frac{1}{2}, t} + \frac{\Delta t \cdot q_{i,t} \cdot q_{i-\frac{1}{2}, t}} {2 \cdot \theta_{i-\frac{1}{2}, t}} \right ) \cdot \frac {c_{i-1, t} - c_{i, t}} {(\Delta z)^2}\]

\[- \left ( \theta_{i+\frac{1}{2}, t} \cdot D_{i+\frac{1}{2}, t} - \frac{\Delta z}{2} \cdot q_{i+\frac{1}{2}, t} + \frac{\Delta t \cdot q_{i,t} \cdot q_{i+\frac{1}{2},t}} {2 \cdot \theta_{i+\frac{1}{2},t}} \right ) \cdot \frac{c_{i,t} - c_{i+1,t}}{(\Delta z)^2}\]

\[- \frac{q_{i+\frac{1}{2}, t} \cdot c_{i,t} - q_{i-\frac{1}{2}, t} \cdot c_{i-1,t} }{\Delta z}\]

and for \(q1/2 < 0\):

\[\frac{ \theta_{i , t+\Delta t} \cdot c_{i, t+\Delta t} - \theta_{i, t} \cdot c_{i, t}} {\Delta t} =\]

\[\left ( \theta_{i - \frac{1}{2}, t} \cdot D_{i-\frac{1}{2}, t} - \frac{\Delta z}{2} \cdot q_{i-\frac{1}{2}, t} + \frac{\Delta t \cdot q_{i,t} \cdot q_{i-\frac{1}{2}, t}} {2 \cdot \theta_{i-\frac{1}{2}, t}} \right ) \cdot \frac {c_{i-1, t} - c_{i, t}} {(\Delta z)^2}\]

\[- \left ( \theta_{i+\frac{1}{2}, t} \cdot D_{i+\frac{1}{2}, t} - \frac{\Delta z}{2} \cdot q_{i+\frac{1}{2}, t} + \frac{\Delta t \cdot q_{i,t} \cdot q_{i+\frac{1}{2},t}} {2 \cdot \theta_{i+\frac{1}{2},t}} \right ) \cdot \frac{c_{i,t} - c_{i+1,t}}{(\Delta z)^2}\]

\[- \frac{q_{i+\frac{1}{2}, t} \cdot c_{i+1,t} - q_{i-\frac{1}{2}, t} \cdot c_{i,t} }{\Delta z}\]

\(\theta\) Volumetric water content \([m^3 \, m^{-3}]\)
\(c\) Solute concentration \([kg \, m^{-3}]\)
\(D\) Effective dispersion coefficient \([m^2 \, d^{-1}]\)
\(q\) Water flux density \([m \, d^{-1}]\)
\(i\) Soil layer
\(z\) Soil depth \([m]\)
\(t\) Time \([d]\)

In case of high water fluxes, the time step is reduced for numerical stability reasons.

\(t=1.0\) for q < 5.0 mm d–1 \([d]\)
\(t=0.5\) for q > 5.0 mm d–1 und q < 10.0 mm d–1 \([d]\)
\(t=0.25\) for q > 10.0 mm d––1 und q < 15.0 mm d–1 \([d]\)
\(t=0.125\) for q > 15.0 mm d–1 \([d]\)