Stability Bias in Lagrangian (Back)tracking in Divergent Flows - Reijnders, Daan; Denes, Michael C.; Rühs, Siren; Breivik, Øyvind; Nordam, Tor; van Sebille, Erik (John Wiley and Sons Inc., 2026)
Forward- and backward-in-time Lagrangian advection, used to determine fate and origin of material in the ocean, are mathematically consistent. However, their numerical computations are hampered by round-off and truncation errors. Trajectory calculations are stable to errors (i.e., errors are dampened) in zones of velocity convergence and unstable (errors are amplified) in regions of divergence. The stability to errors thus flips when time integration is reversed, which, depending on the numerical configuration, can lead to significant discrepancies between forward- and backward-in-time trajectories. As divergence statistics can be asymmetrical and may be inhomogeneously distributed in space, this can lead to what we call the “stability bias.” Using representative numerical set-ups, we show that already for timescales of less than half a year, there can be systematic basin-scale biases in which regions are identified as particle origins or sinks. While the stability bias is linked to divergence, it is not only limited to 2D trajectories in 3D flows, as we discuss how inappropriate treatment of surface boundary conditions in 3D Lagrangian studies can also introduce an effective non-zero divergence. Backtracking is typically applied to material that has accumulated in convergent zones, for which the stability bias especially impedes source attribution studies. Furthermore, we show how discrepancies between forward and backward trajectories can make a Bayesian approach to backtracking unsuitable. We advise modelers to routinely compare forward- and backward trajectories and assess the bias in different numerical set-ups to increase study robustness. Analytical integration methods are less error-prone and may be preferred over RK4.
Keywords: Lagrangian; backtracking; numerical integration; trajectories; Global and Planetary Change; Environmental Chemistry; General Earth and Planetary Sciences