Transit-Time and Tracer-Age Distributions in Geophysical Flows

Abstract

Transport in the atmosphere and in the ocean is the result of the complex action of time-dependent and often highly turbulent flow. A useful diagnostic that summarizes the rate at which fluid elements are transported from some region to a point (or the reverse) via a multiplicity of pathways and mechanisms is the probability density function (pdf) of transit times. The first moment of this pdf, often referred to as ?mean age,? has become an important transport diagnostic commonly used by the observational community. This paper explores how to probe the flow with passive tracers to extract transit-time pdf?s. As a foundation, the literal ?tracer age? is defined as the elapsed time since tracer was injected into the flow, and the corresponding tracer-age distribution, Z, as the fractional tracer mass in a given interval of tracer age. The distribution, Z, has concrete physical interpretation for arbitrary sources, but is only equivalent to a tracer-independent transit-time pdf of the flow in special cases. The transit-time pdf is a propagator, G?, of boundary conditions (the ?age spectrum? of T. M. Hall and R. A. Plumb) applied over a control surface, Ω. The propagator G? is shown to be the flux into Ω resulting from a unit mass injected into the time-reversed flow. Through explicit construction of the transit-time pdf using the concept of tracer age, the special cases for which Z and G? coincide are established. This allows a direct physical demonstration of G?, and its adjoint G??, as the pdf?s of transit times since fluid at point r had last contact with Ω, and until fluid at r will have first contact with Ω, respectively. In the limit as Ω is shrunk to a point, point-to-point transit-time pdf?s are well defined, but their mean transit time and higher-order moments become infinite. Several concrete geophysical examples are considered to illustrate under what conditions characteristics of tracer-age and transit-time pdf?s can be inferred from observations in the atmosphere or the ocean.