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

Abstract

AbstractBased 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 diffusivityand local effective diffusivity, to quantify localized, instantaneous, irreversible mixing. The attractiveness of these two diagnostics are that 1) they both recovers exactly the effective diffusivityKeffproposed 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. WhileKeffcannot 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.