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

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-ww(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) = Sc-1/2, where Sc is the molecular Schmidt number and eta is the Kolmogorov microscale defining the smallest scale of turbulent eddies impacted by fluid viscosity. Using an approximate solution to the von Karman-Howarth equation predicting D-ww(r) in the inertial and viscous regimes, prior formulations for k(L) are recovered including (i) kL = root 2/15Sc(-1/2)v(K), v(K) is the Kolmogorov velocity defined by the Reynolds number v(K)eta/nu = 1 and nu is the kinematic viscosity of water; (ii) surface divergence formulations; (iii) k(L) alpha Sc(-1/2)u(*), where u(*) is the waterside friction velocity; (iv) k(L) alpha Sc-1/2 root g nu/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) = root 2/15Sc(-1/2) (nu g beta(o)q(o))(1/4) in free convection, where q(o) is the surface heat flux and beta(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.