Testing the Assumptions Underlying Ocean Mixing Methodologies using Direct Numerical Simulations

Small-scale turbulence plays an important role in setting the large-scale properties and circulation of the ocean. There has been a concerted, and highly valuable effort for many decades to take direct measurements in the world's oceans to infer key eddy diffusivities, using a variety of methods. The results of direct numerical simulations of stratified turbulence are used to test several fundamental assumptions involved in three different methods commonly used to estimate turbulent diffusivity from field measurements. When calculated using the volume-averaged dissipation rates from the simulations, the vertical eddy diffusivities of heat inferred from the `Osborn' method (based around turbulent dissipation rate $\epsilon$) and `Osborn-Cox' method (based around temperature variance destruction rate) are within 40\% of the value diagnosed using the volume-averaged buoyancy flux for all cases, while the Thorpe scale method (based around overturning scales) performs similarly well in a simulation with a relatively large buoyancy Reynolds number but significantly overestimates the vertical diffusivity in simulations with $Re_b \equiv \epsilon/\nu N^2<60$, where $N$ is the characteristic buoyancy frequency, and $\nu$ is the kinematic viscosity. A limited number of vertical profiles randomly selected from the computational volume are also considered to gain insight into confidence intervals for inevitably finite observational data sets. The Osborn, Osborn-Cox and Thorpe scale methods converge to their respective estimates based on volume-averaged statistics faster than the vertical diffusivity calculated directly from the buoyancy flux, which is contaminated with reversible contributions from internal waves. When applied to a small number of vertical profiles, several assumptions underlying the Osborn and Osborn-Cox methods are not well-supported by the simulation data. However, the vertical diffusivity inferred from these methods compares reasonably well to the exact value from the simulations and outperforms the assumptions underlying these methods in terms of the relative error. Consistently with recent theoretical developments, the Osborn method might well provide a reasonable approximation to the diffusivity associated with the irreversible, buoyancy flux.