Tracer age symmetry in advective–diffusive flows

The “age” of a trace constituent is a common diagnostic of its transport in a geophysical flow. Deleersnijder et al. (Bull. Soc. R. Sci. Liege 70 (2001a) 5) and Beckers et al. (SIAM J. Appl. Math. 61 (2001) 1526) analyzed tracers released from point sources in unbounded advective–diffusive flows with uniform coefficients and noted a surprising feature: the “mean tracer age” (the averaged elapsed time since tracer was injected) is symmetric about the source, despite the directionality of the flow. Although the majority of tracer is swept downstream, the small fraction that diffuses upstream does so at the same average rate. We explore this symmetry physically by examining the random walk trajectories that underlie the advective–diffusive description of transport. Using physical arguments, we show that symmetry in the tracer age field is a natural consequence of symmetry in the velocity and diffusivity fields.