In a recent paper, we derived an analytical expression for the deposition velocity, v_{d}, of molecular hydrogen on soil that includes the action of a dry top soil layer without H_{2} removal (Ehhalt and Rohrer, 2013). This expression is based on the solution of the vertical diffusion equation in a two-layer model and takes the following form:
The two-layer model was first suggested by Yonemura et al. (2000), and it assumes uniform conditions in the respective layers. δ is the depth of the dry top layer, D_{S} is the diffusivity of H_{2} in the soil (D_{S,I} in the dry top layer, D_{S,II} in the moist, deeper soil layer), k_{s} is the rate constant for removal of H_{2} from soil air, and Θ_{a} is the fraction of soil volume filled with air. Expressions for D_{S}, δ, k_{s} Θ_{a}, that is, their dependences on soil moisture, Θ_{w}, and temperature, T, are also given in the earlier paper (Ehhalt and Rohrer, 2013). Thus, eq. (1) can be used to describe the dependence of v_{d} on Θ_{w} and T. This description is more complete and more realistic than that derived from the one-layer model used so far (Yonemura et al., 2000; Smith-Downey et al., 2008; Morfopoulos et al., 2012).
Yet, eq. (1) does not treat all situations. In particular, it does not include the impact on v_{d} of a production of H_{2} within the soil. Such H_{2} production has been shown, for instance, to accompany the fixation of nitrogen by bacteria (Conrad and Seiler, 1980). Further evidence of H_{2} production within the soil is provided by the occasional observation of non-zero asymptotic H_{2} mixing ratios at greater soil depths (cf. Smith-Downey et al., 2008).
With this short note, we would like to point out that eq. (1) can be easily expanded to include a production of H_{2} within the soil. This expansion is given by the factor (1 −M_{e}/M_{a}), such that the deposition velocity, v_{d,p}, including soil production is given as
where v_{d} is given by eq. (1). M_{a} is the H_{2} mixing ratio in the atmosphere and M_{e}=P/(k_{s} Θ_{a}·ρ) is the equilibrium mixing ratio established in the soil between the production with the rate P and the destruction k_{s} Θ_{a}. ρ is the number density of air. For v_{d} derived from the one-layer model, this relation has already been shown to hold (Yonemura et al., 2000). In Appendix A, we show that it also holds for the two-layer model.
Clearly, the dry deposition velocity defined by eq. (2) is no longer independent of the atmospheric mixing ratio of H_{2}. This has implications for the geographical distribution of v_{d,p} which is especially important when the global uptake of H_{2} by soil is derived from inverse modelling. We further note that M_{e} can be obtained from the field measurement of v_{d,p} by the chamber method by allowing the H_{2} mixing ratio in the chamber to drop to its asymptotic value (cf. Conrad and Seiler, 1985; Rice et al., 2011).
By definition, the flux of H_{2} from the atmosphere into the soil is given by
The top soil layer, layer I, is assumed to be so dry that neither bacterial destruction nor production of H_{2} can take place. Thus, throughout layer I the vertical H_{2} flux, F_{I}, remains constant and equal to F_{a}. The gradient in M_{S}(z), the mixing ratio in the soil, is therefore linear and
For z ≥ δ, that is, in layer II, the vertical profile of M_{S}(z) is given by the one-dimensional vertical diffusion equation
To determine the free parameter M_{0}, we calculate the H_{2} flux into layer II, F_{II} (δ), and use the fact that at the immediate boundary the flux into layer II is identical to the flux in layer I. The flux into layer II is given by Fick's law
Inserting eq. (A.4) for M_{S,II}(z) yields
The H_{2} mixing ratio at depth δ, M_{S}(δ), is then
Inserting eq. (A.9) into eq. (A.2) and remembering that F_{I}=F_{a}, we obtain
Collecting the terms with F_{a} gives
Conrad R. , Seiler W . Contribution of hydrogen production by biological nitrogen fixation to the global hydrogen budget . J. Geophys. Res . 1980 ; 85 : 5493 – 5498 .
Conrad R. , Seiler W . Influence of temperature, moisture, and organic carbon on the flux of H_{2} and CO between soil and atmosphere: field studies in subtropical regions . J. Geophys. Res . 1985 ; 90 : 5699 – 5709 .
Ehhalt D. H. , Rohrer F . Deposition velocity of H_{2}: a new algorithm for its dependence on soil moisture and temperature . Tellus B . 2013 ; 65
Morfopoulos C. , Foster P. N. , Friedlingstein P. , Bousquet P. , Prentice I. C . A global model for the uptake of atmospheric hydrogen by soils . Global Biogeochem. Cycles . 2012 ; 26 : GB3013 .
Rice A. , Dayalu A. , Quay P. , Gammon R . Isotopic fractionation of atmospheric hydrogen . Biogeosciences . 2011 ; 8 : 763 – 769 .
Smith-Downey N. V. , Randerson J. T. , Eiler J. M . Molecular hydrogen uptake by soils in forest, desert, and marsh ecosystems in California . J. Geophys. Res . 2008 ; 113 : G03037 .
Yonemura S. , Yokozawa M. , Kawashima S. , Tsuruta H . Model analysis of the influence of gas diffusivity in soil on CO and H_{2} uptake . Tellus B . 2000 ; 52 : 919 – 933 .