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

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  • 1 aState Key Laboratory of Tropical Oceanography, South China Sea Institute of Oceanology, Chinese Academy of Sciences, Guangzhou, China
  • | 2 bSouthern Marine Science and Engineering Guangdong Laboratory (Guangzhou), Guangzhou, China
  • | 3 cGuangxi Key Laboratory of Marine Disaster in the Beibu Gulf, Bubei Gulf University, Qinzhou, China
  • | 4 dState Key Laboratory of Satellite Ocean Environment Dynamics, Second Institute of Oceanography, Ministry of Natural Resources, Hangzhou, Zhejiang, China
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Abstract

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 Keff proposed by 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 Keff 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 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.

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.

© 2022 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Shiqiu Peng, speng@scsio.ac.cn

Abstract

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 Keff proposed by 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 Keff 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 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.

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.

© 2022 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Shiqiu Peng, speng@scsio.ac.cn
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