Modelling eddy diffusivity in the ocean with scalar-velocity cospectra

Abstract. This paper outlines a pathway to understanding eddy diffusivity in stratified flows by considering scalar-velocity cospectra, also called scalar flux spectra (E_wθ). Its main objective is to unify existing scalings of eddy diffusivity into a quantitative recipe. We construct a synthetic model scalar-velocity cospectrum for stratified turbulence applicable to the ocean. Its integral over all wavenumbers, normalized by the mean scalar gradient, equals eddy diffusivity and scales principally with horizontal Froude number Fr_h and buoyancy Reynolds number Re_b. Our model spectrum reproduces published mixing coefficient estimates from oceanic microstructure and direct numerical simulation (DNS), as well as a new analysis of existing eddy covariance data, over the entire accessible range of Fr_h, or equivalently, Ozmidov/Thorpe length scale ratios. However, scaling between eddy diffusivities and Fr_h for strongly stratified flows, Fr_h < 1, appears to differ between most DNS and oceanic measurements. We particularly emphasize the importance of the spectral slope s_b between the Ozmidov and integral length scales, finding best agreement with ocean observations when assuming s_b ≈ −4/3, but s_b ≈ 1/3 for DNS for the stratified wavenumber range. We argue that this difference may be due to the length scale of the forcing. From direct measurements of scalar-velocity cospectra in the ice-ocean boundary layer, we find that additional contributions to eddy diffusivity at low Froude numbers may result from background processes not scaling with Fr_h or Re_b, which warrants further investigation into boundary layer processes.