A Structure Function Model Recovers the Many Formulations for Air‐Water Gas Transfer Velocity

Two ideas regarding the structure of turbulence near a clear air‐water interface are used to derive a waterside gas transfer velocity kLfor sparingly and slightly soluble gases. The first is that kLis proportional to the turnover velocity described by the vertical velocity structure function Dww(r), where ris separation distance between two points. The second is that the scalar exchange between the air‐water interface and the waterside turbulence can be suitably described by a length scale proportional to the Batchelor scale lB=ηSc−1/2, where Scis the molecular Schmidt number and ηis the Kolmogorov microscale defining the smallest scale of turbulent eddies impacted by fluid viscosity. Using an approximate solution to the von Kármán‐Howarth equation predicting Dww(r) in the inertial and viscous regimes, prior formulations for kLare recovered including (i) kL=2/15Sc−1/2vK, vKis the Kolmogorov velocity defined by the Reynolds number vKη/ν= 1 and νis the kinematic viscosity of water; (ii) surface divergence formulations; (iii) kL∝Sc−1/2u∗, where u∗is the waterside friction velocity; (iv) kL∝Sc−1/2gν/u∗for Keulegan numbers exceeding a threshold needed for long‐wave generation, where the proportionality constant varies with wave age, gis the gravitational acceleration; and (v) kL=2/15Sc−1/2(νgβoqo)1/4in free convection, where qois the surface heat flux and βois the thermal expansion of water. The work demonstrates that the aforementioned kLformulations can be recovered from a single structure function model derived for locally homogeneous and isotropic turbulence. The problem considered here is a theoretical prediction of mass transfer across and air‐water interface. This interfacial transfer phenomenon is featured prominently in global carbon balances, methane, nitrous oxides, dimethyl sulfide, and other gases. It is used to assess the metabolic health of aquatic ecosystems and to determine evasion rates of volatile organic compounds from lakes, estuaries, reservoirs, and large water treatment plants. The novelty of the approach is to link a bulk quantity reflecting the efficiency of swirling motions near the air‐water interface to the sizes of eddies and their energetic content responsible for the aforementioned swirling motion. The proposed approach is shown to recover a number of equations describing gas transport across interfaces that summarize a large corpus of experiments and simulations. Scaling laws empirically describing gas transfer velocity kLacross marine and coastal systems are theoretically explainedNew theory predicts kLusing Kolmogorov's universal structure function scaling laws for turbulence instead of surface renewal theoryWork shows how multiple mechanisms with different assumptions subject to eddy‐turnover time constraint lead to similar scaling laws for kL