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 K-L and local effective diffusivity K-eff, to quantify localized, instantaneous, irreversible mixing. The attractiveness of these two diagnostics is that 1) they both recover 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 time step and hence a local equivalence between particle-based and tracer-based diffusivities can be obtained instantaneously. From a particle perspective, K-L represents the local magnifying of the mixing length; from a contour perspective, K-eff 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, K-L 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, K-L & nbsp;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, K-eff 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.& nbsp;SIGNIFICANCE STATEMENT:: Large discrepancies are reported in the estimates of mixing using different types of diffusivity diagnostics, specifically the particle-based, tracer-based, and Eulerian-based diffusivities, as their definitions are quite different from each other. Here we propose two local mixing diagnostics according to the particle- and tracer-based diffusivities. It is then shown that the theoretical discrepancies between the three types of mixing diagnostics can be clearly reconciled based on these two local diagnostics. Therefore, an updated, consistent, and unified view of different mixing models becomes clear and discrepancies between different estimates can be minimized.