The Lagrangian particle method is widely used to understand scalar tracer concentra- tion 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 ad- vective component is captured by traditional Lagrangian approaches. In this work, we report a novel method that closes tracer budgets on Lagrangian trajectories in a man- ner consistent with Eulerian budgets in finite-volume models. The scalar tracer concen- trations on grid cell walls are derived from the model advection scheme and then inter- polated inside grid boxes along streamlines. The divergence of the diffusive flux and re- action terms are interpolated based on velocity and tracer concentration, ensuring the tracer budget closes in terms of both trajectory and volume integrals. 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 analy- ses 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.

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 naturally-occurring and hypothetical 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 paleooceanography, including future opportunities.