Gas Transfer Across Air-Water Interfaces in Inland Waters: From Micro-Eddies to Super-Statistics

In inland water covering lakes, reservoirs, and ponds, the gas exchange of slightly soluble gases such as carbon dioxide, dimethyl sulfide, methane, or oxygen across a clean and nearly flat air-water interface is routinely described using a water-side mean gas transfer velocity $\overline{k_L}$, where overline indicates time or ensemble averaging. The micro-eddy surface renewal model predicts $\overline{k_L}={\alpha }_{o}S{c}^{-1/2}{\left(\nu \overline{ϵ}\right)}^{1/4}$, where $Sc$ is the molecular Schmidt number, $\nu$ is the water kinematic viscosity, and $\overline{ϵ}$ is the waterside mean turbulent kinetic energy dissipation rate at or near the interface. While ${\alpha }_o=0.39-0.46$ has been reported across a number of data sets, others report large scatter or variability around this value range. It is shown here that this scatter can be partly explained by high temporal variability in instantaneous $ϵ$ around $\overline{ϵ}$, a mechanism that was not previously considered. As the coefficient of variation $\left(C{V}_e\right)$ in $ϵ$ increases, ${\alpha }_o$ must be adjusted by a multiplier ${\left(1+C{V}_e^2\right)}^{-3/32}$ that was derived from a log-normal model for the probability density function of $ϵ$. Reported variations in ${\alpha }_o$ with a macro-scale Reynolds number can also be partly attributed to intermittency effects in $ϵ$. Such intermittency is characterized by the long-range (i.e., power-law decay) spatial auto-correlation function of $ϵ$. That ${\alpha }_o$ varies with a macro-scale Reynolds number does not necessarily violate the micro-eddy model. Instead, it points to a coordination between the macro- and micro-scales arising from the transfer of energy across scales in the energy cascade.