Quantifying Local, Instantaneous, Irreversible Mixing Using Lagrangian Particles and Tracer Contours

Based on the dispersion of Lagrangian particles relative to the contours of a quasi-conservative tracer field, the present study proposes two new diffusivity diagnostics: the local Lagrangian diffusivity and local effective diffusivity , to quantify localized, instantaneous, irreversible mixing. The attractiveness of these two diagnostics are that 1) they both recovers exactly the effective diffusivity K eff proposed by Nakamura (1996) when averaged along a contour and 2) they share very similar spatial patterns at each timestep and hence a local equivalence between particle-based and tracer-based diffusivities can be obtained instantaneously. From particle perspective, represents the local magnifying of the mixing length; from contour perspective, represents the local strengthening of tracer gradient and elongation of the contour interface. Both of these enhancements are relative to an unstirred (meridionally sorted) state. While K eff cannot quantify the along-contour variation of irreversible mixing, is able to identify the portion of a (quasi-conservative) contour where it is leaky and thus easily penetrated through by Lagrangian particles. Also, unlike traditional Lagrangian diffusivity, is able to capture the fine-scale spatial structure of mixing. These two new diagnostics allows one to explore the interrelations among three types (Eulerian, Lagrangian, and tracer-based) of mixing diagnostics. Through a time mean, has a very similar expression with the Eulerian Osborn-Cox diffusivity. The main difference lies in the definition of their denominators. That is, the non-eddying tracer background state, representing the lowest mixing efficiency, differs in each definition. Discrepancies between these three types of diffusivities are then reconciled both theoretically and practically.