Tracer Budgets on Lagrangian Trajectories

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.