On the symmetry of the age field of a passive tracer released into a one-dimensional fluid flow by a point-source
Institut d'astronomie et de géophysique G. Lemaitre (ASTR), Université catholique de Louvain, 2 Chemin du Cyclotron, B-1348 Louvain-la-Neuve, Belgium, ericd@astr.ucl.ac.be
Institut de mathématique, Sart Tilman B37, Université de Liège, B-4000,Liège, Belgium, E.Delhez@ulg.ac.be.
Institut d'astronomie et de géophysique G. Lemaitre (ASTR), Université catholique de Louvain, 2 Chemin du Cyclotron, B-1348 Louvain-la-Neuve, Belgium, crucifix@astr.ucl.ac.be
GeoHydrodynamics and Environment Research (GHER), Sart Tilman B5, Université de Liège, B-4000 Liege, Belgium, jm.beckers@ulg.ac.be
Abstract
Tracer is released from a point-source into an incompressible, one-dimensional fluid flow, with constant velocity and diffusivity. The age of a tracer parcel, which is defined as the time elapsed since leaving the source, may be evaluated as the ratio of the age concentration to the tracer concentration. The latter are governed by two partial differential equations. Time-dependent analytical solutions are derived, which show that the age is symmetric with respect to the source. This is astonishing, since it could have been expected that the age would reflect somehow the strong asymmetry of the tracer concentration, which tends to be much larger on the downstream side of the source than on the upstream side. Some finite-difference counterparts of this problem are seen to lead to age fields which, in their steady-state limit, are also symmetric with respect to the source. This is believed to be helpful to interpret the results of numerical models of complex fluid flows in which the age is introduced as a diagnostic variable.