Symmetry and asymmetry of water ages in a one-dimensional flow

Hall and Haine [J. Mar. Syst., in press] briefly addressed the problem of estimating the age of irreducible fluid elements or water parcels in a one-dimensional flow with constant velocity and diffusivity. Herein further developments are achieved on this subject. The age of every water parcel is set to zero at the moment it passes through the point x=0, where x is an appropriate space coordinate. As time progresses, the age of the water is seen to increase unboundedly upstream of the point x=0, and tend to the steady-state value x/u downstream of the point x=0, where u is the water velocity. By contrast, the age of the water parcels that have touched at least once the point x=0 is symmetric with respect to the point x=0 is smaller than the water age at any time and position, and tends to x/u as time progresses. Asymptotic expansions are derived for large times.