## 2 Materials and Methods
### Calculating Fluxes Using the Gradient Flux Method  
#### 2.2.1 Theoretical Background  
Soil gas monitoring provides data like soil gas concentration that can be interpreted directly, e.g., elevated concentrations of N\({}_{2}\)O or C\({}_{2}\)H\({}_{4}\) that indicate ongoing denitrification or elevated activity of certain microbes. If soil gas concentrations are linked with their exact sampling position and the transport conditions in the soil, gas flux and gas production can be calculated using the gradient \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Gas** & **GC** & **Column** & **Detector** & \begin{tabular}{c} **Limit of** \\ **Detection** \\ \end{tabular} & **Precision** & \begin{tabular}{c} **Atmospheric** \\ **Concentration** \\ \end{tabular} & \begin{tabular}{c} **Typical Soil** \\ **Air * \\ \end{tabular} \\ \hline CO\({}_{2}\) & & & CP-PoraBond Q & & 30 ppm & 1.0\% & ca. 400 ppm & \
\begin{tabular}{c} 500–20,000 \\ ppm \\ \end{tabular} \\ \cline{3-6} O\({}_{2}\) & & & TCD & 0.5\% & 0.6\% & 20.95\% & 0–20.95\% \\ N\({}_{2}\) & Perkin Elmer & CP-Molsieve 5A & & 1.7\% & 0.6\% & 78.08\% & ca. 78.08\% \\ Ar & Clarus 680 GC & & & 0.02\% & 0.8\% & 0.93\% & ca. 0.93\% \\ \hline CH\({}_{4}\) & & & & 0.27 ppm & 1.6\% & 1.95 ppm &
\begin{tabular}{c} \textless{}0.3 to \\ \textgreater{}100 ppm \\ \end{tabular} \\ C\({}_{2}\)H\({}_{4}\) & & & CP-SilicaPlot & FID & 0.04 ppm & 2.3\% at & 0 ppm & 0 to \textgreater{}5 ppm \\ N\({}_{2}\)O & & & CP-PoraBond Q & ECD & 0.02 ppm & 1.1\% & 0.33 ppm & \textless{}0.2–10 ppm \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Set-up and analytical performance of the GC systems used. Relative precision of measurement is given at ambient concentration unless stated otherwise. method [44; 50] (Figure 2). Using the gradient method implies that molecular diffusion is the dominating gas transport process in the soil. This means other gas transport processed like mass flow of air [79] and turbulence induced pressure-pumping [80; 81] need to be ruled out. These processes can play temporarily a role in highly porous soils and high wind speed, yet their average contribution is probably small. Gas diffuses in all directions, which would require a three-dimensional mathematical consideration of gas diffusion in soils [16; 22]. In most cases, however, a reduction of the soil profile to a one-dimensional representative simplification is justified and common practices in soil chemistry, soil physics and hydraulic modelling. Hence, gas transport in the soil is often approximated by [PERSON]'s first law, which is reduced to its one-dimensional formulation. \[F=-D_{\mathrm{S}}\rho_{air}\frac{\partial\chi_{C}}{\partial z}, \tag{1}\]  
where \(F\) is the gas flux (mol m\({}^{-2}\) s\({}^{-1}\)) at the depth \(z\) (m), \(D_{\mathrm{S}}\) the effective gas diffusion coefficient of the gas species in the soil (m\({}^{2}\) s\({}^{-1}\)), \(\rho_{air}\) the mean air density (\(\upmu\)mol m\({}^{-3}\)), \(\chi_{C}\) the molar fraction (e.g., atmospheric CO\({}_{2}\) = \(\chi_{CO_{2}}\) = 440 ppm or 440 \(\upmu\)mol mol\({}^{-1}\)) of the diffusing gas species at depth \(z\). It is this gradient in molar fraction which drives molecular diffusion, and not the gradient in molar density ( \(C_{\mathrm{CO_{2}}}\); \(\upmu\)mol m\({}^{-3}\)) [44; 82]. The net production of a gas in soil profile must suffice the conservation of matter, that means:  
\[\frac{\partial C_{\mathrm{sum}}}{\partial t}=-\frac{\partial F}{\partial z}+P, \tag{2}\]  
where \(C_{\mathrm{sum}}\) is the amount of gas molecules per soil volume (\(\upmu\)mol m\({}^{-3}\)), \(t\) is time (s), and \(P\) the gas production (\(\upmu\)mol m\({}^{-3}\) s\({}^{-1}\)) Theoretically, \(C_{\mathrm{sum}}\) includes the amount of gas in air-filled pores and dissolved in the soil solution, since gas molecules exchange between these compartments. Practically, the dissolution of most trace gases is negligible, if the dissolution of CO\({}_{2}\) in a carbonated soil solution is excluded [65; 83].