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The Lagrangian particle method is widely used to understand scalar tracer concentration fields in models of the atmosphere and oceans. Simulating virtual particles provides an alternative description of advection to the Eulerian representation in models and aids in identifying pathways, timescales, and connectivity. Atmospheric and oceanic models solve advection‐diffusion‐reaction equations to simulate tracers, in which only the advective component is captured by traditional Lagrangian approaches. In this work, we report a novel method that closes tracer budgets on Lagrangian trajectories in a manner consistent with Eulerian budgets in finite‐volume models. The scalar tracer concentrations on grid cell walls are derived from the model advection scheme and then interpolated inside grid boxes along streamlines. The divergence of the diffusive flux and reaction terms are interpolated based on velocity and tracer concentration, ensuring the tracer budget closes in terms of both trajectory and volume integrals. Compared to the Eulerian budget analysis, which considers a fixed volume, our method quantifies the tracer evolution within a volume that moves along with the flow. We demonstrate the method using a case study of Southern Ocean biogeochemistry. Another case study involves analyzing the heat budget of the 2011 Western Australian marine heat wave. The method bridges the gap between Eulerian budget and Lagrangian particle analyses by representing the advective processes with particle movements and interpolating the diffusive and reactive processes onto trajectories in a way consistent with the finite‐volume description. 

{"Abstract":["Released to complement a revised version of Haine et al. 2024.\n\nJulia package to complement "A Review of Green's Function Methods in Ocean Circulation Models," by Haine et al. (2024). See https://essopenarchive.org/users/528978/articles/1215807-a-review-of-green-s-function-methods-for-tracer-timescales-and-pathways-in-ocean-models. This package has two goals.\n\n1. One of the stated goals of the manuscript is to make Green's Function methods accessible for learning purposes. A computational notebook is provided with this package and detailed below.\n\n2. Here, we also aim to make a Julia package that is useful and computationally efficient for research purposes. The codes are contained in the source directory (`src`) and can be used and imported by other Julia projects. \n\n "]} 

Arguably, the most conspicuous evidence for anthropogenic climate change lies in the Arctic Ocean. For example, the summer-time Arctic sea ice extent has declined over the last 40 years and the Arctic Ocean freshwater storage has increased over the last 30 years. Coupled climate models project that this extra freshwater will pass Greenland to enter the sub-polar North Atlantic Ocean (SPNA) in the coming decades. Coupled climate models also project that the Atlantic Meridional Overturning Circulation (AMOC) will weaken in the twenty-first century, associated with SPNA buoyancy increases. Yet, it remains unclear when the Arctic anthropogenic freshening signal will be detected in the SPNA, or what form the signal will take. Therefore, this article reviews and synthesizes the state of knowledge on Arctic Ocean and SPNA salinity variations and their causes. This article focuses on the export processes in data-constrained ocean circulation model hindcasts. One challenge is to quantify and understand the relative importance of different competing processes. This article also discusses the prospects to detect the emergence of Arctic anthropogenic freshening and the likely impacts on the AMOC. For this issue, the challenge is to distinguish anthropogenic signals from natural variability. This article is part of a discussion meeting issue ‘Atlantic overturning: new observations and challenges’. 

Abstract Understanding advective‐diffusive dispersal of trace substances in environmental fluids like the global ocean is a ubiquitous challenge in geophysics. Since the turn of the millennium, substantial progress has been made in the theory, implementation in models, and application of such tracers in oceanography. For the first time, this progress is reviewed here in a synthetic way. We focus on tracer techniques in ocean models, including real and virtual tracers that diagnose timescale information, and we emphasize the connection to the Green's function that solves the advection‐diffusion equation. Implementation of these techniques in ocean models is explained in an accessible way. We present example applications of these techniques to questions concerning ocean circulation, transport of biogeochemicals, and paleoceanography, including future opportunities.