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# Dry deposition of molecular hydrogen in the presence of H2 production

## Abstract

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How to Cite: Ehhalt, D.H. and Rohrer, F., 2013. Dry deposition of molecular hydrogen in the presence of H2 production. Tellus B: Chemical and Physical Meteorology, 65(1), p.20620. DOI: http://doi.org/10.3402/tellusb.v65i0.20620
Published on 01 Jan 2013
Accepted on 13 Apr 2013            Submitted on 13 Feb 2013

In a recent paper, we derived an analytical expression for the deposition velocity, vd, of molecular hydrogen on soil that includes the action of a dry top soil layer without H2 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:

(1 )
${\text{v}}_{\text{d}}=\frac{\text{1}}{\frac{\delta }{{\text{D}}_{\text{S},\text{I}}}+\sqrt{\frac{1}{{\text{D}}_{\text{S},\text{II}}\cdot {\text{k}}_{\text{s}}{\Theta }_{\text{a}}}}}$

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, DS is the diffusivity of H2 in the soil (DS,I in the dry top layer, DS,II in the moist, deeper soil layer), ks is the rate constant for removal of H2 from soil air, and Θa is the fraction of soil volume filled with air. Expressions for DS, δ, ks Θa, that is, their dependences on soil moisture, Θw, 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 vd on Θw 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 vd of a production of H2 within the soil. Such H2 production has been shown, for instance, to accompany the fixation of nitrogen by bacteria (Conrad and Seiler, 1980). Further evidence of H2 production within the soil is provided by the occasional observation of non-zero asymptotic H2 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 H2 within the soil. This expansion is given by the factor (1 −Me/Ma), such that the deposition velocity, vd,p, including soil production is given as

(2 )
${\text{V}}_{\text{d},\text{p}}={\text{V}}_{\text{d}}\cdot \left(1-{\text{M}}_{\text{c}}/{\text{M}}_{\text{a}}\right),$

where vd is given by eq. (1). Ma is the H2 mixing ratio in the atmosphere and Me=P/(ks Θa·ρ) is the equilibrium mixing ratio established in the soil between the production with the rate P and the destruction ks Θa. ρ is the number density of air. For vd 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 H2. This has implications for the geographical distribution of vd,p which is especially important when the global uptake of H2 by soil is derived from inverse modelling. We further note that Me can be obtained from the field measurement of vd,p by the chamber method by allowing the H2 mixing ratio in the chamber to drop to its asymptotic value (cf. Conrad and Seiler, 1985; Rice et al., 2011).

## Appendix A

### Derivation of vd,p

By definition, the flux of H2 from the atmosphere into the soil is given by

(A.1 )
${\text{F}}_{\text{a}}={\text{V}}_{\text{d},\text{p}}\cdot \rho \cdot {\text{M}}_{\text{a}},$
where Fa is given in units of molec cm−2 s−1, vd,p, the dry deposition velocity in the presence of H2 production, has the units of cm s−1, ρ, the number density of air is in molec cm−3, and Ma the atmospheric mixing ratio has the units of molec molec−1.

The top soil layer, layer I, is assumed to be so dry that neither bacterial destruction nor production of H2 can take place. Thus, throughout layer I the vertical H2 flux, FI, remains constant and equal to Fa. The gradient in MS(z), the mixing ratio in the soil, is therefore linear and

(A.2 )
${\text{F}}_{\text{I}}=\rho \cdot {\text{D}}_{\text{S},\text{I}}\cdot \frac{{\text{M}}_{\text{S}}\left(\text{0}\right)-{\text{M}}_{\text{S}}\left(\delta \right)}{\delta }$
Here, MS(0)=Ma and M(δ) are the H2 mixing ratios at depth z=0 (the surface) and at z=δ, the depth of the dry layer. DS,I is the diffusivity in layer I, its units are cm2 s−1.

For z ≥ δ, that is, in layer II, the vertical profile of MS(z) is given by the one-dimensional vertical diffusion equation

(A.3 )
${\Theta }_{\text{a}}\cdot \rho \cdot \frac{\partial {\text{M}}_{\text{S}}}{\partial \text{t}}=\frac{\partial }{\partial \text{z}}\cdot \rho \cdot {\text{D}}_{\text{S},\text{II}}\cdot \frac{\partial {\text{M}}_{\text{S}}\left(\text{z}\right)}{\partial \text{z}}-\rho \cdot {\text{M}}_{\text{S}}\left(\text{z}\right)\cdot {\text{k}}_{\text{s}}{\mathrm{\Theta }}_{\text{a}}+\text{P}$
where DS,II is the diffusivity in the deeper soil layer II, units of cm2 s−1, P is the production rate of H2 per soil volume in molec cm−3 s−1, ks is the rate constant for the removal of H2 from soil air, units of s−1, and Θa is the fraction of soil volume filled with air. Assuming steady state, that is, MS/t=0 and DS,II, ρ, Θa, ks, P to be constant with depth equation (A.3) can be solved analytically:
(A.4 )
${\text{M}}_{\text{S},\text{II}}\left(\text{z}\right)={\text{M}}_{\text{0}}\cdot \text{exp}\left(\frac{-\left(\text{z}-\delta \right)}{\zeta }\right)+{\text{M}}_{\text{e}}$
where the characteristic decay length $\zeta =\sqrt{{\text{D}}_{\text{S},\text{II}}/{\text{k}}_{\text{s}}{\mathrm{\Theta }}_{\text{a}}}$ and Me=P/(ksΘa·ρ) is the equilibrium H2 mixing ratio established between in soil production and destruction of H2 in the absence of transport.

To determine the free parameter M0, we calculate the H2 flux into layer II, FII (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

(A.5 )
${\text{F}}_{\text{II}}\left(\delta \right)=-\rho \cdot {\text{D}}_{\text{S},\text{II}}\cdot {\frac{\partial {\text{M}}_{\text{S},\text{II}}\left(\text{z}\right)}{\partial \text{z}}\mid }_{\text{z}=\delta }$

Inserting eq. (A.4) for MS,II(z) yields

(A.6 )
${\text{F}}_{\text{II}}\left(\delta \right)=\rho \cdot {\text{D}}_{\text{S},\text{II}}\cdot {\text{M}}_{0}/\zeta .$
Remembering that Fa=FI=FII(δ) we can rewrite eq. (A.6)
(A.7 )
${\text{F}}_{\text{a}}=\rho \cdot {\text{D}}_{\text{S},\text{II}}\cdot {\text{M}}_{0}/\zeta .$
(A.8 )
$\text{Or}{\text{M}}_{0}=\left({\text{F}}_{\text{a}}\cdot \zeta \right)/\left(\rho \cdot {\text{D}}_{\text{S},\text{II}}\right).$

The H2 mixing ratio at depth δ, MS(δ), is then

(A.9 )
${\text{M}}_{\text{s}}\left(\delta \right)={\text{M}}_{0}+{\text{M}}_{\text{e}}=\left({\text{F}}_{\text{a}}\cdot \zeta \right)/\left(\rho \cdot {\text{D}}_{\text{S},\text{II}}\right)+{\text{M}}_{\text{e}}.$

Inserting eq. (A.9) into eq. (A.2) and remembering that FI=Fa, we obtain

(A.10 )
${\text{F}}_{\text{a}}=\frac{\rho \cdot {\text{D}}_{\text{S},\text{I}}}{\delta }\cdot \left({\text{M}}_{\text{a}}-{\text{M}}_{\text{e}}-\frac{{\text{F}}_{\text{a}}\cdot \zeta }{\rho \cdot {\text{D}}_{\text{S},\text{II}}}\right)$

Collecting the terms with Fa gives

(A.11 )
${\text{F}}_{\text{a}}\cdot \left(\frac{\delta }{{\text{D}}_{\text{S},\text{I}}}+\frac{\zeta }{{\text{D}}_{\text{S},\text{II}}}\right)=\rho \cdot \left({\text{M}}_{\text{a}}-{\text{M}}_{\text{e}}\right),$
and since Fa=vd,p · ρ · Ma [see eq. (A.1)]
(A.12 )
${\text{v}}_{\text{d},\text{P}}=\left(\text{1}-\frac{{\text{M}}_{\text{e}}}{{\text{M}}_{\text{a}}}\right)\cdot \left(\frac{\text{1}}{\frac{\delta }{{\text{D}}_{\text{S},\text{I}}}+\frac{\zeta }{{\text{D}}_{\text{S},\text{II}}}}\right)$
or
(A.13 )
${\text{v}}_{\text{d},\text{P}}=\left(\text{1}-\frac{{\text{M}}_{\text{e}}}{{\text{M}}_{\text{a}}}\right)\cdot {\text{v}}_{\text{d}}$
since the second term on the right-hand side of eq. (A.12) is identical to vd from eq. (1).

## References

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