A Structure Function Model Recovers the Many Formulations for Air‐Water Gas Transfer Velocity. Issue 9 (3rd September 2018) - Katul, Gabriel; Mammarella, Ivan; Grönholm, Tiia; Vesala, Timo
Abstract: Two ideas regarding the structure of turbulence near a clear air‐water interface are used to derive a waterside gas transfer velocity k L for sparingly and slightly soluble gases. The first is that k L is proportional to the turnover velocity described by the vertical velocity structure function D w w ( r ), where r is 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 l B = η S c −1/2, where S c is the molecular Schmidt number and η is 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 D w w ( r ) in the inertial and viscous regimes, prior formulations for k L are recovered including (i) k L = 2 / 15 S c − 1 / 2 v K, v K is the Kolmogorov velocity defined by the Reynolds number v K η / ν = 1 and ν is the kinematic viscosity of water; (ii) surface divergence formulations; (iii) k L ∝ S c − 1 / 2 u ∗, where u ∗ is the waterside friction velocity; (iv) k L ∝ S c − 1 / 2 gν / u ∗ for Keulegan numbers exceeding a threshold needed for long‐wave generation, where the proportionality constant varies with wave age, g is the gravitational acceleration; and (v) k L = 2 / 15 S c − 1 / 2 ( νg β o q o ) 1 / 4 in free convection, where q o is the surface heat flux and β o is the thermal expansion of water. The work demonstrates that the aforementioned k L formulations can be recovered from a single structure function model derived for locally homogeneous and isotropic turbulence. Plain Language Summary: 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. Key Points: Scaling laws empirically describing gas transfer velocity k L across marine and coastal systems are theoretically explained New theory predicts k L using Kolmogorov's universal structure function scaling laws for turbulence instead of surface renewal theory Work shows how multiple mechanisms with different assumptions subject to eddy‐turnover time constraint lead to similar scaling laws for k L