Universal Scaling Law for Gas Transfer Velocities across Complex Interfaces

The significance of turbulent gas exchanges near interfaces (water or solid) on human health, ecosystem and climate-related processes is not in dispute. A massive degassing of CO2 that killed more than 1200 people in August of 1986 from Lake Nyos (Cameroon), the hotly debated total CO2 emissions from inland waters (now estimated around 2 Pg C per year), the rapid uptick in Nitrous Oxide emissions from wastewater treatment plants and farming are a small subset from numerous examples prominently featured in news reports and science plans alike. While the significance of such mass transport at water or rough-surface interfaces is not in dispute, the mechanisms describing its efficiency by turbulence remains multifaceted and subject to active research. There is now a renaissance in mass transfer studies across interfaces because of rapid advancements in (i) measurements (e.g. remote sensing using high resolution infrared cameras, fiber optic cables, stable isotopic techniques, routine eddy-covariance turbulent flux measurements), and (ii) simulations (e.g. Large Eddy Simulations and Direct Numerical Simulations). What is lagging (or perhaps even lacking) are theoretical (even phenomenological) developments describing the efficiency of this turbulent transport process. Instead of dealing with fluxes and concentration, it is preferable to deal with so-called gas transfer velocity (=k) when quantifying turbulent transport efficiency. This k is defined as the mass flux at the interface normalized by the concentration difference between the interface and the bulk flow away from the interface. Operational formulations for k, especially the waterside k widely used in climate models are based on mean air velocity (=U) at a reference height of 10 m above the water surface. Similar formulations are in use above rough solid surfaces as may be encountered in evaporation studies from bare soil surfaces when the water vapor source is not limiting the evaporation rate. The goal of this talk is to explicitly bring the spectrum of eddies into the description of k using micro-meteorological theories. In particular, two ideas regarding the structure of turbulence near a clear air-water interface are used to derive a waterside gas transfer velocity k for sparingly and slightly soluble gases (similar derivation can be adopted for gas transfer from rough surfaces). The first is that k is proportional to the turnover velocity described by the vertical velocity structure function D_ww(r), where r is effective eddy size contributing to the interfacial transfer. The second is that the scalar exchange between the air-water interface and the waterside turbulence can be suitably described by a length scale r proportional to the so-called Batchelor scale (related to the Kolmogorov microscale via a molecular Schmidt number). Using an approximate solution to the von Kármán-Howarth equation predicting the shape of D_ww(r) in the inertial and viscous regimes, prior expressions for k that encompass a large corpus of experiments are recovered. The approach covered here deviates from other theoretical approaches that are based on surface renewal theories, where the contact times between eddies and the interface and/or the fractional source area must be specified. The debate about the role of large eddies, especially at a lower Reynolds number R* formed from an integral length scale and root-mean squared velocity is also featured. The so-called 'chain saw model' where large eddies carry small eddies to the free surface is used to interpret their role. In this chain-saw model, turbulent exchange of scalars resembles a chain saw cutting through a wood section. The small teeth on the chain saw (i.e. eddies of size r commensurate to the Batchelor scale) perform the actual cutting, the larger teeth (i.e. inertial scales) remove wood chips but do not participate in the actual cutting, and the overall rotational speed of the chain saw itself is dictated by external conditions such as the motor speed setting (i.e. large scale vorticity set by boundary conditions where the coherent vortical structures are eddy sizes commensurate to the integral scale of the flow). Provided the large scales do not interact and distort the micro-scales (to be expected at high R* as evidenced by a number of studies dealing with the so-called random sweeping decorrelation hypothesis, a micrometeorological theory dealing with the scaling laws of higher-order spectra), it is shown that the micro-eddy model result derived here and from surface renewal remains satisfactory (i.e. k proportional to the Kolmogorov velocity). The impact of thermal stratification and a formulation for k in the limit of free convection is also presented.