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 kL for sparingly and slightly soluble gases. The first is that kL is proportional to the turnover velocity described by the vertical velocity structure function Dww(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 lB=ηSc−1/2, where Sc 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 Dww(r) in the inertial and viscous regimes, prior formulations for kL are recovered including (i) kL=2false/15Sc−1false/2vK, vK is the Kolmogorov velocity defined by the Reynolds number vKη/ν = 1 and ν is the kinematic viscosity of water; (ii) surface divergence formulations; (iii) kL∝Sc−1false/2u∗, where u∗ is the waterside friction velocity; (iv) kL∝Sc−1false/2gνfalse/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) kL=2false/15Sc−1false/2false(νgβoqofalse)1false/4 in free convection, where qo is the surface heat flux and βo is the thermal expansion of water. The work demonstrates that the aforementioned kL formulations can be recovered from a single structure function model derived for locally homogeneous and isotropic turbulence.

Climate‐Sensitive Controls on Large Spring Emissions of CH4 and CO2 From Northern Lakes

“…Because darkened floating chambers could have biased daytime CO 2 emissions by limiting photosynthetic uptake, we instead estimated CO 2 fluxes using pCO 2 from water samples (2015–2017) and a boundary layer model: where k is the gas transfer velocity and C aq − C air , eq represents the concentration difference between the water in the boundary layer and at equilibrium with the air above, determined from weekly measurements of surface water CO 2 concentrations (Figure ) and assuming an atmospheric mixing ratio of 400 ppm. We determined k with a surface renewal model (Lamont & Scott, ): where c 1 is a scaling parameter determined to be ∽0.4 on theoretical grounds (Katul et al, ; Lamont & Scott, ) and from analysis of multiple experiments including physical measurements of turbulence and eddy covariance (Zappa et al, ), ν is the kinematic viscosity, Sc is the Schmidt number for CO 2 (Jähne et al, ), and ε is the dissipation rate of turbulent kinetic energy, which is driven by wind shear ( u * w ) and the buoyancy flux under cooling ( β ) (Tedford et al, ): …”

“…where c 1 is a scaling parameter determined to be ∽0.4 on theoretical grounds (Katul et al, 2018;Lamont & Scott, 1970) and from analysis of multiple experiments including physical measurements of turbulence and eddy covariance (Zappa et al, 2007), ν is the kinematic viscosity, Sc is the Schmidt number for CO 2 (Jähne et al, 1987), and ε is the dissipation rate of turbulent kinetic energy, which is driven by wind shear (u *w ) and the buoyancy flux under cooling (β) (Tedford et al, 2014):…”

Gas exchange in streams and rivers

“…Given that gas flux (F) is the product of a gas exchange velocity and concentration gradient, for example, Equation 1, much of the variation in estimates of F stem from variation in k. A large and old body of theory guides understanding of controls on k (Bennett & Rathbun, 1972), and this theory is advancing today (e.g., Katul, Mammarella, Grönholm, & Vesala, 2018). Here we provide a general overview.…”

A primer on turbulence in hydrology and hydraulics: The power of dimensional analysis

“…The results suggest that the ‘−1/4’ scaling law is an outcome of the Kolmogorov microscale and the associated time scale, while the viscous corrections only alter the scaling of F E with respect to Sc but not u * (G. Katul & Liu, ). Revisions to surface renewal theories that account for large eddies have been proposed and reviewed elsewhere (G. Katul, Mammarella, Grönholm, & Vesala, ). These theories confirm that the similarity constant A E in Equation 39 vary as power‐law with a Reynolds number, a finding that has been confirmed by DNS and several laboratory studies (G. Katul et al, ).…”