## 1. Introduction

Trace gases are an important part of the atmosphere not only because their radiative and chemical properties affect climate but also because they directly probe atmospheric transport. Routine observations of winds do not sample the atmosphere sufficiently to resolve many mechanisms important for global transport, such as convection, tropopause folds, breaking gravity waves, and boundary-layer mixing. Measurements of tracers provide one of the best ways to quantify transport rates and to assess the realism of transport in atmospheric models. To this end, considerable effort continues to be expended on measurements of chemically long-lived tracers from ground-based stations, aircraft, balloon, and satellite platforms (e.g., Bischof et al. 1985; Luo et al. 1994; Maiss et al. 1996; Boering et al. 1996; Geller et al. 1997; Elkins et al. 1996; Mote et al. 1998). However, direct and unambiguous determination of transport properties from tracer data is often difficult or impossible because the sources and sinks of tracers are imperfectly known. Even assuming perfect knowledge of sources and sinks, transport information is intricately entangled with their spatial and temporal variability. Therefore, a diagnostic framework connecting tracer distributions to tracer-independent transport properties of the flow is a valuable tool for the interpretation of observations.

One component of such a framework is what Hall and Plumb (1994, hereafter HP94) called the “age spectrum.” The age spectrum, **r**. More precisely, **r** transit times. In this paper, we therefore refer to the age spectrum as the transit-time pdf and reserve “tracer-age distribution” for a distinct quantity defined below. The pdf **r** pathways available to the fluid. Transit-time pdf’s have been discussed and applied to stratospheric transport analysis by Kida (1983) and by HP94. HP94 defined the age spectrum as the distribution of times since the fluid elements constituting a given stratospheric air parcel had last contact with the troposphere. The mean transit time, Γ, at a point in the stratosphere (“mean age” in the language of HP94) is the average time since the air there was last in the troposphere. The transit-time pdf has proved to be a useful conceptual tool for interpreting tracer observations in the stratosphere. Although

HP94 formulated the transit-time pdf in terms of boundary conditions on mixing ratio by analyzing the stratospheric response to a short-lived impulse in mixing ratio imposed near the earth’s surface. This is a natural approach for the stratosphere, into which most air and tracer enters through the tropical tropopause regardless of its history or source–sink distribution in the troposphere. In many situations, however, it is more physical to consider tracer distributions as arising from specified sources and sinks rather than from imposed mixing ratios at the surface. For example, the distribution of CO_{2} in the troposphere, including near the surface, is usually considered to be determined by the combination of atmospheric transport and surface sources and sinks, rather than transport and imposed surface mixing ratio.

In this paper we establish a novel conceptual and analytical framework that connects the transit-time pdf with a closely related, but generally distinct, distribution of tracer age. The tracer-age distribution, *Z,* keeps track of the time for which tracer particles have been in the flow, and is defined here as the fraction of current tracer mass binned according to the time (i.e., age) it has been in the flow. As we shall show, consideration of explicit sources and sinks rather than mixing ratio boundary conditions leads naturally to the tracer-age distribution, *Z,* and mean tracer age, *A,* which are complimentary to *Z* and *Z* and *last* had contact with a specified region, Ω. Our approach, however, is general and also includes the pdf of times for an air parcel to have *first* contact with Ω. We illustrate these concepts with examples from both simple analytic models and from numerical atmospheric transport models, and we demonstrate how characteristics of *Z* can be inferred from real tracers. We also explore the dependence of the transit-time pdf on the size of Ω and find the general result for advecting-diffusing systems that mean transit times become infinite as Ω is shrunk to a point.

While the atmosphere provides the context and language for this paper, the development has more general applicability. Embedding the transit-time pdf in the general framework of Green functions provides a guide for synthesizing measurements of tracer into a coherent, physical picture of transport, without regard to the particular geophysical setting. Oceanographers commonly describe oceanic tracer transport in terms of “age,” generally construed as the mean time since a water parcel last had contact with the ocean surface (e.g., Jenkins 1987; England 1995), and Beining and Roether (1996) discuss age distributions for ocean parcels, a concept related to the pdf’s developed here. Following examples for the atmosphere, we briefly discuss application of our approach to the ocean. Advecting-diffusing systems are in fact so common in nature that the applicability of these concepts extends to nongeophysical fields, such as migrating biological populations (e.g., Zabel and Anderson 1997), though we will address only geophysical issues here.

## 2. Green functions and boundary propagator

We now introduce Green functions for passive tracer transport and develop their connection to a propagator of boundary conditions (BCs) on mixing ratio, which has the interpretation of a transit-time pdf. Details of the general solution for mixing ratio in terms of Green functions are provided in appendix A. The development in this section is general, but throughout we provide illustrations from simple analytical models whose details are provided in appendixes B and C. In section 5 examples from more realistic general circulation models (GCMs) are given. Additional examples of Green functions in a GCM context may by found in Holzer (1999).

### a. Green functions

*χ.*We write the tracer continuity equation as

_{t}

*χ*

*S,*

*S*represents a specified source of tracer. Although formally it does not matter whether the diffusive part of

*χ*to be synthesized as a linear superposition of the mixing ratios from individual source pulses. We, therefore, consider

*S*to be a collection of pulses localized in space and time, which in the limit become Dirac

*δ*functions. Equation (1) with

*S*being one such

*δ*function defines the corresponding Green function,

*G,*that is,

_{t}

*G*

**r**

*t*

**r**

*t*

*ρ*

^{−1}

*δ*

*t*

*t*

*δ*

^{3}

**r**

**r**

*ρ*is the density of the fluid. Thus,

*G*has dimensions of inverse mass and corresponds to tracer injected at the point (

**r**′,

*t*′) normalized by the mass injected. Because there cannot be any tracer in the fluid before it has been injected,

*G*is said to satisfy the causality condition that

*G*= 0 for

*t*′ >

*t.*The solutions of (1) and (2) depend on a consistent set of BCs.

*χ*

**v**

**∇***χ*

*ρ*

^{−1}

**∇***ρκ*

**∇***χ*

**v**(

**r**,

*t*), and isotropic (eddy-) diffusivity,

*κ*(

**r**,

*t*). (Generalization to anisotropic diffusivity is straightforward, but not considered explicitly here.) Denoting the Green function satisfying generic BCs by

*G*

_{x}, the general solution of (1) with (3) is (see appendix A)

**r**

_{s}on the boundary, ∂, of the geophysical reservoir under consideration (e.g., the earth’s surface for the atmosphere). The first term of (4) represents the time-evolved initial condition

*χ*(

**r**′, 0). The second term is the superposition of all the tracer pulses emitted at points

**r**′ and times

*t*′ by the source

*S*(

**r**′,

*t*′). The last term is an integral over the boundary (outward normal

**n̂**) and represents any contributions to

*χ*due to the BCs imposed on

*χ*and/or

**n̂**·

**∇***χ.*In addition, there may be an independently specified BC on the normal flow,

**n̂**·

**v**. We will consider only two specific sets of BCs, discussed next.

#### 1) Zero-flux boundary conditions

*G*to denote the Green function subject to zero-flux BCs. With these BCs, the boundary term of (4) vanishes entirely and the general solution to (1) becomes

*dm*′ ≡

*ρ*(

**r**′,

*t*′)

*d*

^{3}

*r*′. Note that this case includes specified surface fluxes as these can be considered as interior sources or sinks placed arbitrarily close to the boundary. [Alternatively,

**n̂**·

**∇***χ*could be specified as a flux BC, which would lead to an additional boundary contribution given by the first term of (5). However, for our purposes it is more convenient to regard specified surface fluxes as sources and use (6).]

Because zero-flux BCs are imposed on *G,* the tracer “mass,” ∫ *dm* *G* = 1, is conserved. Ultimately, the initial unit tracer mass injected is uniformly spread over the entire reservoir so that *G* has the constant long-term, spatially uniform limit of *G*_{∞}, where *G*_{∞} = ∫ *dm* *G*/∫ *dm* = 1/*M*_{A} is the inverse mass of the fluid (assumed constant).

Figure 1 shows *G* for the simple case of a one-dimensional (1D) diffusive atmosphere (exponentially decaying fluid density; for details see appendix B). Because this model’s transport is stationary, *G* depends on time only through the elapsed time (or “lag time”) *t* − *t*′ ≡ *ξ.* For the *G* shown, the source is located at the bottom boundary (*z*′ = 0). Note that for long times, *G* approaches a constant *G*_{∞} = 1, and that *G* overshoots *G*_{∞} at locations, *z,* close to the source. The overshoot is a generic feature of *G* (Holzer 1999).

We emphasize that closed-form analytical solutions for Green functions are rare exceptions and only possible for very simple cases. In the general case, *G* is physically a “puff” of time-evolving mixing ratio that typically becomes rapidly shredded and filamented by the time-evolving velocity field. The complicated dependence of *G* on the flow is manifest when expressing *G* as a (Feynman-type) path integral (see, e.g., Shraiman and Siggia 1994), though we will not make use of path integrals in this paper.

#### 2) Boundary conditions on mixing ratio

*G*

_{0}, so that from (4)

**∇**_{rs}

**r**

_{s}on Ω. The purely diffusive nature of the boundary term in (7) is general and simply due to the BC

*G*

_{0}= 0 on Ω.

Because of the zero-mixing-ratio BCs on *G*_{0} over Ω (denoted as *χ*0BC), *G*_{0} leaks continuously out of Ω so that *G*_{0} has the long-time limit of zero. Note that if the source point, **r**′, lies on Ω, then *G*_{0}(**r**, *t*|**r**′, *t*′) = 0 at all points **r**. This is a statement of the fact that as the source point approaches Ω, mass is lost at an increasing rate, until a source right on Ω looses all its mass instantaneously.

Figure 2 shows *G*_{0} as a function of elapsed time, *t* − *t*′ ≡ *ξ,* for the 1D atmosphere of appendix B. Zero-mixing-ratio BCs are imposed at the bottom boundary (i.e., Ω corresponds to *z* = 0). Note that *G*_{0} → 0 as *ξ* → ∞ and that the evolution of *G*_{0} depends on the proximity of the field point, *z,* to both the source point, *z*′, and to Ω. The largest response, *G*_{0}, to the initial unit mass injection is seen at the source location. Close to Ω, the *χ*0BCs exert a stronger influence resulting in a smaller amplitude of *G*_{0}. The most slowly decaying response can be seen at the field point most remote from Ω.

### b. Boundary propagator and transit-time pdf

*G*′, that makes use of the fact that a BC on mixing ratio specified on the surface, Ω, can be represented as a sum of

*δ*functions in time and surface location. By contrast

*G*and

*G*

_{0}were based on a

*δ*function decomposition of the sources. Thus,

*G*′ is directly defined through a

*δ*function BC without any explicit sources, that is,

*G*′ satisfies

_{t}

*G*

**r**

*t*

**r**

_{0}

*t*

*G*

**r**

_{s}

*t*

**r**

_{0}

*t*

*δ*

*t*

*t*

*δ*

^{2}

**r**

_{s}

**r**

_{0}

**r**

_{s}and

**r**

_{0}are points on Ω. The solution to (1) resulting from the general BC,

*χ*(

**r**

_{0},

*t*), in the absence of explicit sources, is then given by

*G*′ “propagates” the BC on

*χ*from Ω into the interior of the domain. Note that causality demands here that

*G*′(

**r**,

*t*|

**r**

_{0},

*t*′) = 0 for

*t*′ >

*t.*

*G*′ follows from (10). Consider the case

*χ*(

**r**

_{0},

*t*) = ϒ(

*t*−

*t*

_{0}), where the Heaviside function ϒ(

*t*−

*t*

_{0}) is unity for

*t*>

*t*

_{0}and zero otherwise. In that case, after waiting for an infinitely long time (

*t*

_{0}→ −∞), the boundary value of unity will have propagated throughout the domain, and if the domain is finite (as is any reasonable geophysical reservoir) then

*χ*(

**r**,

*t*) = 1, giving the normalization

**r**transit times. (Note that HP94 use the symbol

*G*for the age spectrum.) As we will show below,

*dt*′

**r**,

*t*|Ω,

*t*′) is the probability that a fluid element at (

**r**,

*t*) had last contact with Ω at a time between

*ξ*≡

*t*−

*t*′ and

*ξ*+

*dξ*ago.

Note that a propagator, *G*′, for general BCs on mixing ratio over an extended surface, Ω, such as the earth’s surface, must be defined as in (8) and (9). Boundary propagators with the *δ*(*t* − *t*′) BC applied only to a small subregion of Ω, and zero-flux BCs elsewhere on Ω, cannot be used to synthesize the response to an Ω-distributed BC on mixing ratio from a superposition of such propagators. The *G*′ only combine appropriately via (10) when the BC (9) applies over the entire control surface, Ω.

Figure 3 illustrates the general character of *t* − *t*′ ≡ *ξ* for the 1D atmosphere of appendix B. Close to Ω (in this case, *z* = 0), most fluid elements have experienced a short transit time since they were last at Ω, and *z,* from Ω the most probable time since last contact with Ω (where

### c. Probabilistic interpretations

For a physical and probabilistic interpretation of the Green functions it is useful to have a concept of the entity that is being transported. To this end, we envision the fluid as consisting of the standard “material fluid elements” of fluid mechanics. A subtlety arises when we consider a tracer. Generally tracer molecules that label a material fluid element do not remain confined to that fluid element but rather diffuse to adjacent fluid elements by molecular diffusion. However, we are interested here in the limiting case where the molecular diffusivity is negligible compared to the turbulent diffusivity (large Péclet number). Therefore, we regard any diffusion in the transport operator,

The Green functions *G* and *G*_{0} have the probabilistic interpretation of being pdf’s so that *G* *dm* is the probability of finding a tracer-marked particle in the fluid mass element *dm* at time *t,* if that particle was located at **r**′ at time *t*′, and similarly for *G*_{0}. Correspondingly, the domain-integrated masses, *M* ≡ ∫ *dm* *G* = 1 and *M*_{0}(*t*|**r**′, *t*′) ≡ ∫ *dm* *G*_{0}(**r**, *t*|**r**′, *t*′), have the interpretation of being the probabilities that a marked particle released at (**r**′, *t*′) is still marked a time *t* − *t*′ later. In the case of *G* (zero-flux BCs), once the particle has been marked with tracer, it will remain marked forever and is, therefore, to be found somewhere with unit probability. In the case of *G*_{0} (zero-mixing-ratio BCs), the marked particle will eventually make contact with the *χ*0BC control surface, where it will loose its tracer marker, so that *M*_{0} approaches zero for long *t* − *t*′. The decaying probability *M*_{0} is shown in Fig. 4 for the 1D atmosphere of appendix B with Ω being the “surface” *z* = 0. Note that the closer the source point, *z*′, is to Ω, the faster *M*_{0} leaks out of the atmosphere.

The “pseudo mass” *M*′(*t*|**r**_{0}, *t*′) ≡ ∫ *dm* *G*′(**r**, *t*|**r**_{0}, *t*′) is not dimensionless and hence not a probability. However, for total fluid mass *M*_{A} = ∫ *dm,* the ratio *P*(*t*|*t*′) ≡ ∫_{Ω} *d*^{2}*r*_{0} *M*′/*M*_{A} = ∫ *dm* *M*_{A} can be interpreted as a pdf. [Note that ^{∞}_{0}*dξ* *P*(*t*|*t* − *ξ*) = 1.] We may think of *P*(*t*|*t*′) as the pdf of “population particle age,” *ξ* ≡ *t* − *t*′, for surface-marked particles which are “born” on Ω and “die” when they make contact with Ω a second time. To see this, rewrite *P*(*t*|*t*′) as ∫ *d*^{3}*r* *μ*(**r**, *t*)**r**, *t*|Ω, *t*′), where *μ*(**r**, *t*) ≡ *ρ*(**r**, *t*)/*M*_{A} is the pdf of finding a particle in the volume *d*^{3}*r.* Given HP94’s interpretation of *μ*(**r**, *t*)**r**, *t*|Ω, *t*′) is the (joint) probability density for finding a particle in the volume *d*^{3}*r* that had last surface contact in the interval (*t*′, *t*′ + *dt*′), and hence has particle age in the interval (*ξ, ξ* + *dξ*). Integrating this joint probability density over **r**, gives the pdf, *P*(*t*|*t*′), for finding in the entire population of particles a particle whose age lies in the interval (*ξ, ξ* + *dξ*). In the following subsection, we confirm HP94’s interpretation of *P*(*t*|*t*′) is singular at *t* = *t*′ for advective-diffusive transport. Figure 5 shows the pdf, *P,* as a function of particle age, *t* − *t*′ ≡ *ξ,* for the 1D atmosphere of appendix B. The pdf *P* diverges like *ξ*^{−1/2} as *ξ* → 0 for this model as indicated on the figure.

### d. *G*′ as flux of *G*_{0} in the time-reversed flow

*G*′ to

*G*

_{0}, which provides important new insight into the interpretation of

*G*′. The boundary propagator

*G*′ is the special case of a

*δ*-function BC on Ω, zero initial conditions, and no explicit sources. Substituting

*χ*(

**r**

_{s},

*t*′) =

*δ*(

*t*′ −

*t*

_{0})

*δ*

^{2}(

**r**

_{s}−

**r**

_{0}) into (7) immediately gives

*G*

**r**

*t*

**r**

_{0}

*t*

_{0}

*ρ*(

**r**

_{0},

*t*

_{0})

*κ*(

**r**

_{0},

*t*

_{0})[

**n̂**·

**∇**

_{r′}

*G*

^{†}

_{0}(

**r**′,

*t*

_{0}|

**r**,

*t*)]

_{r′=r0},

*G*

_{0}in terms of its adjoint

*G*

^{†}

_{0}

*G*

_{0}via the reciprocity condition

*G*

_{0}(

**r**,

*t*|

**r**′,

*t*′) =

*G*

^{†}

_{0}(

**r**′,

*t*′|

**r**,

*t*)

*G*

^{†}

_{0}(

**r**′,

*t*′|

**r**,

*t*)

*t*≥

*t*′, as it does for

*G*

_{0}(

**r**,

*t*|

**r**′,

*t*′). The reciprocity condition is the crucial ingredient in obtaining a general physical interpretation for

*G*′. While

*G*(

**r**,

*t*|

**r**′,

*t*′) takes tracer from (

**r**′,

*t*′) to (

**r**,

*t*) in the flow evolving forward in time,

*G*

^{†}(

**r**′,

*t*′|

**r**,

*t*) takes tracer from (

**r**,

*t*) to (

**r**′,

*t*′) in the time-reversed flow, whose transport operator is

^{†}. Thus,

*G*′(

**r**,

*t*|

**r**

_{0},

*t*

_{0}) has the interpretation of the flux through the control surface, Ω, at (

**r**

_{0},

*t*

_{0}) resulting from a unit-mass injection into the time-reversed flow at (

**r**,

*t*).

It is worth pointing out precisely what “time-reversed” means here. As described in appendix A, to convert the advection–diffusion equation to its adjoint, ∂_{t} → −∂_{t} and **v** · ** ∇** → −

**v**·

**, while**

**∇****· (**

**∇***κρ*

**) remains unchanged. One might at first think that the physical effects of diffusion are, therefore, different. However,**

**∇***G*

^{†}

_{0}(

**r**′,

*t*

_{0}|

**r**,

*t*)

**r**,

*t*) evolving under (

_{t0}

^{†})

*G*

^{†}

_{0}(

**r**′,

*t*

_{0}|

**r**,

*t*)

*δ*

^{3}(

**r**−

**r**′)

*δ*(

*t*−

*t*

_{0})/

*ρ,*with the causality condition that

*t*

_{0}⩽

*t,*so that the time parameter

*t*

_{0}must actually run backward (continually decrease). Thus, the transport in the time-reversed flow represented by

*G*

^{†}

_{0}(

**r**′,

*t*

_{0}|

**r**,

*t*)

**v**→ −

**v**), but diffusion disperses tracer from a point source to a “cloud” just as it does in the time-forward flow.

*G*′, the transit-time pdf

*G*

^{†}

_{0}(

**r**′,

*t*′|

**r**,

*t*)

**r**

*t*

*t*

_{t′}

*M*

^{†}

_{0}(

*t*′|

**r**,

*t*)

*M*

^{†}

_{0}(

*t*′|

**r**,

*t*)

*dm*′

*G*

^{†}

_{0}(

**r**′,

*t*′|

**r**,

*t*)

**r**,

*t*) will still be marked at time

*t*′, with

*t*>

*t*′. Since the flux of probability leaving through Ω at time

*t*is the probability of arriving at Ω between time

*t*and

*t*+

*dt,*

**r**,

*t*) into the time-reversed flow to “first” make contact with Ω. For the time-forward flow,

**r**,

*t*) had

*last*contact with Ω. We shall return to a more direct physical demonstration of the fact that fluxes of

*G*

_{0}into Ω are transit-time distributions when we consider tracer age below.

The relationships (13) and (14) represent one of the main results of this work, and their physical content is summarized schematically in Fig. 6. The pdf of times since fluid at **r** had *last* contact with Ω can be obtained either as the response at **r** to a *δ*-function BC over Ω in the time-forward flow, or in the time-reversed flow as the flux into Ω resulting from a unit mass injection at **r** when *χ*0BCs are imposed on Ω. As will be derived from a more fundamental point of view in section 4, the “adjoint” of this statement is similarly true. The pdf of transit times for fluid at **r** to make *first* contact with Ω can be obtained as either the flux into Ω resulting from a unit-mass injection at **r** under *χ*0BCs in the time-forward flow, or as the response at **r** to a *δ*-function BC over Ω in the time-reversed flow.

*G*′ and

*G*

_{0}can be expressed in another useful form when one writes the gradient as a limit and uses the reciprocity condition and the fact that when

**r**

_{0}is on Ω,

*G*

_{0}(

**r**,

*t*|

**r**

_{0},

*t*′) = 0:

*G*

_{0}is essentially proportional to

*G*′ in the limit as the source point is close to Ω. The closer

**r**

_{0}gets, the more tracer of the initial unit injection of

*G*

_{0}is lost, but this is compensated by scaling

*G*

_{0}with 1/

*ϵ*so that

*G*′ remains normalized as in (11). For the simple 1D atmosphere of appendix B, (15) takes the form

*G*′(

*z, ξ*) = lim

_{z′→0}

*G*

_{0}(

*z, z*′,

*ξ*)/

*z*′ and a plot of

*G*

_{0}/

*z*′ with

*z*′ = 0.001 is indistinguishable from

*G*′ in Fig. 3.

*M*

_{0}and the pdf

*M*′ follows by integrating (13) with respect to

*dm,*giving

*M*

*t*

**r**

_{0}

*t*

_{0}

*ρ*(

**r**

_{0},

*t*

_{0})

*κ*(

**r**

_{0},

*t*

_{0})

**n̂**·

**∇**

_{r′}

*M*

_{0}(

*t*|

**r**′,

*t*

_{0})|

_{r′=r0}.

*M*

_{0}with respect to the source location

**r**′ has consequences for

*M*′. Note that as

*ξ*=

*t*−

*t*′ → 0, the probability,

*M*

_{0}, of finding a marked particle in the atmosphere is unity (for short enough

*ξ*there is no chance of having lost the tracer to the boundary; see also Fig. 4), except for the case where

**r**′ is right on Ω, that is, as

*ξ*→ 0, the probability

*M*

_{0}(

*t*|

**r**′,

*t*−

*ξ*) abruptly goes from unity to zero when

**r**′ reaches Ω. Thus, (16) implies that as

*ξ*→ 0,

*M*′(

*t*

_{0}+

*ξ*|

**r**

_{0},

*t*

_{0}) → ∞ since the gradient of

*M*

_{0}sees a sharp discontinuity at Ω. The boundary propagator

*G*′ has, therefore, an infinite initial pseudo mass,

*M*′, and the population particle-age pdf

*P*(

*t*|

*t*′) = ∫

_{Ω}

*d*

^{2}

**r**

_{0}

*M*′/

*M*

_{A}is also singular at

*t*=

*t*′. This is an expression of the fact that a fluid particle “released” from Ω has overwhelming probability for immediately crashing into Ω again.

We were able to relate *G*_{0} and *G*′ through a simple differential operator because *G*_{0} and *G*′ have compatible BCs. However, *G* and *G*_{0} (and hence *G* and *G*′) obey different BCs and we know of no direct way of obtaining one from the other by applying a differential operator. Nevertheless, *G* and *G*′ can be related by considering either that (a) the evolution of *G* over the control surface, Ω, can be propagated throughout the domain using *G*′; or that (b) *G*′ can be regarded as a “pseudo mixing ratio” whose fluxes through Ω can be used to obtain *G*′ everywhere using *G.* The integral equations connecting *G* and *G*′ follow from the general solution (4) and may be used to derive approximate relationships, but this is not pursued further here.

## 3. Tracers as clocks and tracer-age distributions

We now develop the concept of a distribution of literal tracer age, which has concrete interpretation for arbitrary sources, and which forms the foundation for a direct physical construction of tracer-independent transit-time pdf’s. Fluid marked by tracer naturally contains transit-time information. Consider a parcel at (**r**, *t*) with some tracer mixing ratio. If the source of that tracer was only “on” for a short burst at (**r**′, *t*′), we know that any tracer we find has age *t* − *t*′. This is the basic idea of “tracer age.” Note that this tracer age does not generally date the parcel under consideration as having been at **r**′ at time *t*′ because tracer-marked fluid will generally have been mixed via (turbulent) diffusion with unmarked (“clean”) fluid on the way from **r**′ to **r**. Tracer age is only the transit time of marked particles. Since our interest here lies in using tracers to extract flow characteristics, in section 4 we will explore under what circumstances tracer-age and transit-time distributions coincide.

*χ*0BCs, tracer age is naturally defined as follows. The tracer mixing ratio at the current time,

*t,*is the mixture of tracer mass released at different times,

*t*′, in the past from various locations

**r**′. Imagine giving each fluid particle a clock that is started at the time it is marked with tracer at the source so that the particles marked between time

*t*′ and

*t*′ +

*dt*′ will have “clock time,” or age,

*t*−

*t*′. (For example, one could imagine fluid elements labeled by a radioactive decaying isotope, which acts as a clock.) The mixing ratio of the collection of particles released between time

*t*′ and

*t*′ +

*dt*′ is simply

*dt*′ ∫

*dm*′

*G*

_{x}(

**r**,

*t*|

**r**′,

*t*′)

*S*(

**r**′,

*t*′), where

*G*

_{x}is either

*G*(zero-flux BC) or

*G*

_{0}(

*χ*0BC). Therefore, the fraction of tracer mass residing in a volume

*V*that has been in the flow for time

*t*−

*t*′ is given by

*dt*′

*Z*(

**r**,

*t*|

*S, t*′), where

*V*is given by

*S*(

**r**′,

*t*′) = 0 for

*t*′ <

*t*

_{0}. Note that, by construction, the weighting function,

*Z,*integrates to unity,

^{t}

_{ t0}

*dt*′

*Z*(

*V, t*|

*S, t*′) = 1, so that we may think of

*Z*as the distribution of clock times,

*t*−

*t*′, present in the volume

*V.*We, therefore, refer to

*Z*as a tracer-age distribution, which reduces to a transit-time pdf only under special conditions.

*V*shrinks to a point at

**r**, giving

*dt*′

*Z*is the mass fraction of tracer of age

*t*−

*t*′ that comprises the mixing ratio of the parcel at

**r**. Correspondingly,

**r**.

The formulation of (17) and (19) relied on *χ* being determined through an interior source, *S,* and not through the imposition of any nonzero BCs on mixing ratio. This is not a loss of generality since we can always determine a source–sink field arbitrarily close to the surface that results in the desired surface mixing ratios. (For specified mixing ratios, this requires solving an inverse problem, but for complete surface information, the corresponding implied surface sources are in principle determined.)

As can be seen from (17) and (19), *Z* depends on the space and time dependence of the sources. The character of *Z* also depends strongly on whether zero-flux or zero-mixing-ratio BCs are imposed. Making an analogy with population dynamics, clocks are born at a certain rate at the sources and form a “population” showing various times/ages. Without a “death” process, the population just keeps on aging and mean tracer age increases without bound. This is the case when we impose zero-flux BC (*G*_{x} = *G*); once a clock is born it will tick forever. (As we shall see, the spatial structure of the ever increasing mean tracer age, *A,* carries useful transport information, but *Z* will not approach a stationary state as *t* − *t*′ → ∞.) If *G*_{x} = *G*_{0}, clocks making contact with the *χ*0BC control surface effectively die and the population of clocks reaches a statistically stationary age distribution as *t* − *t*′ → ∞.

## 4. Explicit construction of transit-time pdf’s

To make the connection between the tracer-age distribution, and the pdf of transit times from a point **r**_{A} to some control surface, Ω (which could be shrunk to a point), we consider some idealized experiments. We can straightforwardly obtain the pdf of transit times if we release clocks at a steady rate at **r**_{A}, measure their times as they reach Ω, and then, after measurement, remove the clocks from the system. A normalized histogram of the times recorded is then the pdf of **r**_{A}-to-Ω transit times, denoted here by *Z*_{T}**r**_{A} to any other point.

An idealized tracer experiment to accomplish the desired time keeping feat is constructed as follows: Place a constant source of tracer at **r**_{A} and impose *χ*0BC over Ω to remove clocks after measurement. We must now deal with the subtlety of the fact that right on the surface there is no tracer and hence no clocks. Fortunately, however, we can measure clock times at points **r** close to points **r**_{0} on Ω and take the limit as **r** → **r**_{0}. It is useful to construct our histogram of clock times in terms of the mass fraction residing in a specified volume *V* as discussed in the previous section. We take *V* to be a shell surrounding Ω and then take the limit as the shell has zero thickness followed by the limit of moving this shell onto the surface Ω. The precise result we obtain depends on the details of how we take the limit of the shell collapsing to zero thickness. In order to connect with the results of section 2 relating boundary propagators to fluxes, it is useful to define the shell, *V,* such that its thickness is proportional to the local diffusivity *κ.* In the limit as *V* becomes a thin shell, the integral, ∫_{V} *dm,* then becomes the surface integral, (*δz*/*κ*_{0}) ∫_{∂V} *d*^{2}*r* *ρ*(**r**, *t*)*κ*(**r**, *t*), where *z* is the coordinate normal to the shell surface, *κ*_{0} is an arbitrary constant, and ∂*V* can be taken as the inner surface of *V.* This setup is schematically illustrated in Fig. 7.

*ρ*(

**r**,

*t*)

*S*(

**r**,

*t*) = const. ×

*δ*

^{3}(

**r**−

**r**

_{A}) ϒ(

*t*−

*t*

_{0}) the distribution of clock times for the thin shell,

*V,*is obtained from (17) as

*Z*

_{T}

*t*|

**r**

_{A},

*t*′) ≡ lim

_{V→Ω}

*Z*(

*V, t*|

*S, t*′), that is,

*t*≥

*t*′ ≥

*t*

_{0}. Equation (21) encapsulates a very direct and physical construct for the pdf of

**r**

_{A}-to-Ω transit times. Clocks are born at

**r**

_{A}, the flow takes them to their deaths at Ω, and we make a histogram of their ages just before death. We can now express

*Z*

_{T}

**r**

_{0}on Ω we have

*G*

_{0}(

**r**

_{0},

*t*|

**r**

_{A},

*t*′) = 0, so that with

**r**=

**r**

_{0}−

*ϵ*

**n̂**we have lim

_{ϵ→0}

*G*

_{0}(

**r**

_{0}−

*ϵ*

**n̂**,

*t*|

**r**

_{A},

*t*′)/

*ϵ*=

**n̂**·

**∇**_{r}

*G*

_{0}(

**r**,

*t*|

**r**

_{A},

*t*′)|

_{r=r0}

*ϵ*and taking the limit

*ϵ*→ 0, so that

**r**→

**r**

_{0}, and

*V*→ Ω, we obtain

^{†}(Ω,

*t*|

**r**

_{A},

*t*′) ≡ ∫

_{Ω}

*d*

^{2}

*r*

_{0}

*G*′

^{†}(

**r**

_{0},

*t*|

**r**

_{A},

*t*′), with

*G*

^{†}

**r**

_{0}

*t*

**r**

_{A}

*t*

*ρ*(

**r**

_{0},

*t*)

*κ*(

**r**

_{0},

*t*)

**n̂**·

**∇**

_{r}

*G*

_{0}(

**r**,

*t*|

**r**

_{A},

*t*′)|

_{r=r0},

*G*

_{0}into Ω at

**r**

_{0}.

*Z*

_{T}

*Z*

_{T}

*first contact*with Ω as the normalized net flux into Ω from a unit-mass injection at

**r**

_{A}. Since this flux integrated over all time just gives unity, we can take the limit

*t*

_{0}→ −∞ in (22) to obtain the asymptotic

**r**

_{A}-to-Ω transit-time pdf as

*Z*

_{T}

*t*

**r**

_{A}

*t*

^{†}

*t*

**r**

_{A}

*t*

*t*

*t*

^{†}

*t*

**r**

*t*

_{t}

*M*

_{0}

*t*

**r**

*t*

Note that here *χ*0BC’s are imposed on the destination surface, Ω, and the limit is taken as the *field* point, **r**, approaches Ω. This contrasts to what is involved in the limit (15) relating *G*_{0} to *G**source* point approaches Ω. Also, the interpretation of *last* contact with Ω.

^{†}). We can keep track of transit times in the time-reversed flow through the same construct as in the forward flow. Release clocks with a constant source

*ρ*(

**r**,

*t*)

*S*(

**r**,

*t*) = constant ×

*δ*

^{3}(

**r**−

**r**

_{A}) into the time-reversed flow at

**r**

_{A}, and impose

*χ*0BC over Ω. We construct the distribution of (negative) clock times,

*Z*

_{T}

^{†}(

**r**

_{A},

*t*|Ω,

*t*′)

*G*

^{†}

_{0}

*G*

_{0}and time runs backward. Paralleling the steps that led to (22) using (15) and the reciprocity relation for

*G*

^{†}

_{0}

**r**

_{A}. The time integral of this flux is again normalized [cf. (11) and (12)] so that in the limit

*t*

_{0}→ −∞, we have

*Z*

_{T †}(

**r**

_{A},

*t*|Ω,

*t*′)

**r**

_{A}

*t*

*t*

*t*

*t*

*Z*

_{T†}

*Z*

_{T †}

*last*contact with Ω.

Generally the pdf’s ^{†} are *not* equivalent. A sufficient condition for their equivalence is that *G*_{0}(**r**, *t*|**r**′, *t* − *ξ*) = *G*^{†}_{0}(**r**, −*t*|**r**′,*ξ* − *t*)^{†} can be seen from the following example. Take the case where Ω is a small bubble. Suppose this Ω is connected via a closed stream line to point **r**_{A}, with flow from Ω to **r**_{A} via a short segment of stream line and from **r**_{A} back to Ω via a much longer circuitous segment, and with flow speed roughly constant along the stream line. In this case, the mean time since particles at **r**_{A} had last had contact with Ω is shorter than the mean time for particles leaving **r**_{A} to make first contact with Ω, so that ^{†}.

Consider the limit as Ω is shrunk to a point, say **r**_{B}. In this limit, ^{†} are perfectly well defined as the pdf’s of A-to-B transit times, but their first and higher moments turn out to be infinite. As we have shown, ^{†} are the time-dependent fluxes into Ω resulting from a unit mass injected at **r**_{A} into the time-reversed and time-forward flows, respectively. As the surface area of Ω is decreased, it takes increasingly long for the unit mass to escape through Ω. Because the total mass exiting through Ω is equal to the unit mass initially injected, regardless of the size of Ω, ^{†} remain normalized in the point limit. However, in appendix C section b we show that for 2D and 3D purely diffusive transport, the mass flux out of Ω decays so slowly in the point limit that the first and higher moments of the transit-time pdf’s diverge as Ω vanishes, even for a finite domain. This result generalizes to any advective-diffusive transport because diffusion dominates at small enough spatial scales. Although in the literature one finds point-to-point mean transit times discussed as though they were finite (see, e.g., Plumb and McConalogue 1988; HP94), from the results of appendix C section b, as well as from arguments made in the following section, we conclude that point-to-point mean transit times are infinite.

It is important to note that the divergence of the point A-to-B mean transit time does *not* imply that it will take an infinite time for a unit-mass injection at **r**_{A} to produce a finite mixing ratio (i.e., any finite fraction of *G*_{∞}) at **r**_{B} or any other point, **r**. The characteristic “mixing time,” *τ,* for *χ*(**r**) ∼ *G*_{∞} is finite for a finite reservoir, and represents the time when the majority of marked fluid particles have had a chance to visit **r**. This contrasts with the mean transit time, which is computed as the mean over all particles binned according to their transit time. Because the majority of particles arrive at **r** in a time on the order of *τ,* the transit-time pdf has its peak around *τ* and the divergent moments are, therefore, attributable to slowly decaying tails for large *ξ* = *t* − *t*′ representing “stragglers” taking arbitrarily long to find their point target of Ω (in either the forward or time-reversed flow).

## 5. Transit time information from geophysical tracers

*Z,*and mean tracer age,

*A,*for every point in the domain. Unless

*Z*is computed at special locations under the particular conditions examined in the previous section,

*Z*will not coincide with either

^{†}or

*Z*can still contain information on the mean transit time Γ(

**r**,

*t*) ≡

^{t}

_{−∞}

*dt*′ (

*t*−

*t*′)

**r**,

*t*|Ω,

*t*′) since last contact with the control surface, Ω. One way to see this, is to compare the equations of motion for

*Z*and

*A.*The equation for

*G*′; that is,

_{t}

**r**,

*t*|Ω,

*t*′) =

*δ*(

*t*−

*t*′) for

**r**on Ω. The equation for Γ follows as

_{t}

**r**,

*t*) = 0 for

**r**on Ω, as noted by Boering et al. (1996). Equation (29) has in fact been used to define age (e.g., England 1995).

*Z,*may be written from its definition (19) as

*X*and

*Y,*we defined

*X, Y*) ≡

*XY*) −

*X*

*Y*) −

*Y*

*X*). The BC for

*Z*follows from the BC for the corresponding

*G*

_{x}. The operator

*A*:

*A*and Γ differ through their equations of motion and, generally, also through their BCs. For the case of surface sources,

*S*= 0 in the interior of the domain and (29) and (31) differ through the extra diffusive coupling

*χ*and the BCs.

We now consider several concrete geophysical examples and illustrate some of the concepts developed using numerical atmospheric transport models. Two models are used: the second-generation GCM of the Canadian Climate Centre (CCC) (McFarlane et al. 1992) and an offline chemical transport model (CTM) developed at the Goddard Institute for Space Studies (GISS) (Prather 1986; Prather et al. 1987). The CCC GCM is a spectral model of the troposphere with T32 horizontal resolution and 10 vertical levels to 10 mb. The tracer transport properties of this model have recently been studied in some detail (Holzer 1999). (For the case of surface BCs with spatial discontinuities, spectral transport produces Gibbs oscillations. For plotting only, lowest-level mixing ratios are, therefore, slightly filtered spatially.) The GISS CTM has a grid resolution of 7.8° lat, 10° long, and 21 vertical layers to 0.002 mb (∼90 km) and is driven by a repeated single year of wind and convection data from a version of the GISS GCM with a full stratosphere (Rind et al. 1988). The GISS CTM has been used extensively to study stratospheric tracers (e.g., Hall and Prather 1995). Our purpose here is not to compare results between the two models but rather to illustrate the concepts developed using the model most appropriate for each particular case: CCC GCM for the troposphere and GISS CTM for the stratosphere.

### a. Inferring mean transit times from tracer distributions: Constant surface sources, zero-flux BC

Suppose a tracer with no sinks is injected into the atmosphere with a constant source, that is, with *ρ*(**r**, *t*)*S*(**r**, *t*) = *s*(**r**)ϒ(*t* − *t*_{0}), where *s*(**r**) is constant over a patch, Ω, close to the earth’s surface. The relevant Green function for this problem is *G* (zero-flux BC). Using this source in (20) with *G*_{x} = *G* and taking the long-time limit *G* ∼ *G*_{∞}, we obtain that the mean tracer age *A* ∼ *t*/2. This is intuitive; clocks initialized to zero are injected at a constant rate so that after a long time, *t,* their average clock time reads (1/*t*) ^{t}_{0}*t*′ *dt*′ = *t*/2. Since we are pumping tracer into the atmosphere without any losses, the mixing ratio, *χ,* increases continuously. However, after a long time, *χ* can be decomposed into a linearly growing uniform background, *χ*_{0}, onto which is superposed a statistically stationary, spatially varying, state, *χ*^{+}, that is, *χ* = *χ*_{0} + *χ*^{+}, where *χ*_{0} ≡ *s*_{0}*G*_{∞}*t* with *s*_{0} ≡ ∫ *s*(**r**) *d*^{3}*r* being the total mass injected per unit time.

*A*

^{+}(

**r**,

*t*) ≡

*A*(

**r**,

*t*) −

*t*/2 contains nontrivial transit-time information. In the limit of large

*t, A*

^{+}obeys from (31)

*A*

^{+}has zero-flux BCs. The coupling

*A*

^{+},

*χ*

^{+}) and

*A*

^{+}

*S*are bounded, while

*χ*grows like

*t,*so that

*χ*and

*A*

^{+}

*S*/

*χ*vanish like 1/

*t.*Note that in the interior of the atmosphere, where

*S*= 0, 2

*A*

^{+}and Γ obey the same equation once

*A*

^{+}

*S*/

*χ*and

*χ*are negligible. Recall that for Γ zero-mixing-ratio BCs are applied over Ω, with zero-flux BCs elsewhere. Thus, to the extent that

*A*

^{+}(

**r**,

*t*) ∼ 0 for

**r**on Ω, we can consider 2

*A*

^{+}to obey the same BCs as Γ. It follows that

*A*

**r**

*t*

*A*

^{+}

**r**

*t*

**r**

*t*

*A*(

**r**,

*t*) ≡

*A*(

**r**,

*t*) −

*A*(Ω,

*t*), where

*A*(Ω,

*t*) is the (area-weighted) average of

*A*over Ω.

*A*directly from the definition (20) of

*A,*one needs to know

*G.*Fortunately, however, it is straightforward to show that, in the limit

*t*−

*t*

_{0}→ ∞, the mixing ratio

*χ,*resulting from the constant source, obeys the exact identity

*χ*(

**r**,

*t*) ≡

*χ*(

**r**,

*t*) −

*χ*(Ω,

*t*), where

*χ*(Ω,

*t*) is the average of

*χ*over Ω. One can see this from (20) by taking

*t*−

*t*

_{0}large enough so that the integration range (

*t*

_{0},

*t*) can be split into two parts, (

*t*

_{0},

*t*

_{1}) and (

*t*

_{1},

*t*), where the intermediate time,

*t*

_{1}, is chosen so that

*G*is indistinguishable from

*G*

_{∞}(to within some tolerance) over the interval (

*t*

_{0},

*t*

_{1}). The spatial structure of

*χ*thus comes from the interval (

*t*

_{1},

*t*), where

*G*has temporal and spatial structure. Taking the limit of

*A*(

**r**) −

*A*(

**r**

_{s}) as

*t*

_{0}→ −∞ and averaging

**r**

_{s}over Ω gives (34). From (34) and (33) we have the useful relation

*χ*(

**r**,

*t*) to catch up to

*χ*(Ω,

*t*).

*A*and Δ

*χ*to Γ as in (33) and (35)? We could also have arrived at (35) by using the general boundary propagator,

*G*′. If we assume that

*χ*over Ω is known, we can take it as a time-dependent BC on Ω (with zero flux elsewhere) and use (10)–(12) to obtain in the limit

*t*

_{0}→ −∞

^{∞}

_{0}

*dξ*∫

*d*

^{2}

**r**

_{s}

*G*′

*χ*

^{+}≅

*χ*

^{+}(Ω,

*t*), which is exact only if

*χ*

^{+}is uniform over Ω and time independent corresponding to a strictly uniform, linearly increasing BC over Ω, in which case (35) is obtained as an identity (HP94).

Equation (36) shows that the accuracy of approximation (35) hinges on how uniform *χ*^{+} is over Ω, and how close *χ*^{+} is to being time independent. For constant sources, near-surface mixing ratios can be expected to have a pronounced seasonal cycle, especially in the vicinity of the sources. Thus, it is only in a time average (e.g., over an annual cycle) that the mixing ratio’s growth will be close to linear. However, for the time average (indicated by an overbar) of (36) to reduce to the time average of (35), we also need *G*′*χ*^{+}*G*′*χ*^{+}*G*′ *χ*^{+}*G*′*χ*^{+}*s*_{0}*G*_{∞}Γ, that is, that there is small covariance between the transport operator (represented in integral form by *G*′) and the surface BC. Uniformity over Ω can be ensured by making Ω small enough, but in the limit as Ω is shrunk to a point, Γ and 2Δ*A* will diverge as discussed in section 4.

The general case of a time-dependent, spatially distributed surface source is quite complicated and a full exploration of this case is beyond the scope of this paper. The difficulties associated with spatially distributed sources lie in the fact that there is no unique reference value to define easily interpreted lag times. We note that a spatially uniform source with time variation can be used to extract higher-order moments of the transit-time pdf.

We now illustrate the correspondence between Γ and 2Δ*A* in the troposphere. The CCC GCM was used to simulate the mixing ratio resulting from (a) a constant source applied over a surface patch, Ω, and from (b) a BC on mixing ratio enforced over the same patch that increases linearly in time with rate *γ* (a “ramp” BC). To test our expectation of reasonable agreement between Γ and 2Δ*A* for a small source patch and approximate agreement for extended sources, we performed simulations with two choices of patch size. For the small patch, we chose the 11 grid boxes of the model’s 96 × 48 Gaussian grid within a radius of 790 km of the surface point (50.1°N, 11.3°W) in Europe. This patch has only 0.28% of the global surface area to demonstrate the divergence expected as sources become pointlike. The large patch is defined as the land surface of the Northern Hemisphere, excepting that of Africa and South America. In each case the model is run into quasistationary state, so that mixing ratio increases with the same average rate everywhere in the troposphere. For the constant sources this occurs after a time on the order of the tropospheric mixing time (∼1 yr), but for the ramp-BC case one has to wait for several typical upper tropospheric Γ before the BC is propagated throughout the troposphere, which for the small patch necessitated runs of several decades. For the ramp-BC case, the mean transit time since last contact with the patch is given by Γ = −Δ*χ*/*γ,* and for the constant-source case 2Δ*A* = −Δ*χ*/(*s*_{0}*G*_{∞}). The differential tracer age, 2Δ*A,* has a large seasonal amplitude (not shown) primarily due to the strong seasonality of *A* over the source region. We, therefore, present only annual averages here.

The results for the small patch are shown in Fig. 8. Note that 2Δ*A* is on the order of 15 yr, much larger than the tropospheric mixing time. The shapes of the contours of 2Δ*A* and Γ are virtually identical and their amplitudes agree to within 12%–15%. Given the small patch size, we attribute the differences between 2Δ*A* and Γ to covariance between transport and surface mixing ratio [cf. (36)]. The fields on the lowest model level show the character of the divergence of Γ: steep gradients surround the source region, corresponding to high mixing ratios in that region. Away from the source region, the pattern of Γ is that of the characteristic stationary-state distribution of mixing ratio. We expect that in the limit of vanishing patch size, the “hole,” where Γ drops to zero at the source (corresponding to the peak in mixing ratio), becomes more sharply peaked, driving the characteristic values of Γ to ever higher values, but that the basic pattern of Γ away from the source region remains the same. Because the divergence is localized, it affects the zonal means to a lesser degree and the differences between the Southern Hemisphere values of Γ and those in the vicinity of the source latitude, represent characteristic tropospheric mixing times (Holzer 1999).

In the limit as Ω is shrunk to a point, the correspondence (35) between Γ and 2Δ*A* affords new insight into the nature of the divergence (see appendix Cb). Physically, the cause of the divergence is clear in the point limit: a constant source continually deposits mass into an infinitesimal volume, resulting in infinite mixing ratio at the source, and hence infinite Δ*χ* and Δ*A.*

The relatively good agreement between 2Δ*A* and Γ for the small patch is to be contrasted with the results for the large patch shown in Fig. 9. In the constant source case the flow redistributes near-surface mixing ratio over Ω, so that there are regions where Δ*A* ≡ *A* − *A*(Ω) < 0. The variations of 2Δ*A* over Ω are on the same order as the characteristic tropospheric values of 2Δ*A.* Consequently, 2Δ*A* and Γ have large differences close to Ω and, even in the remote Southern Hemisphere troposphere, the magnitudes of 2Δ*A* and Γ differ by a factor of ∼1.5, although the contour shapes are similar. For this choice of Ω, both 2Δ*A* and Γ are on the order of the tropospheric mixing time as expected, since this time should be on the same order as the time since last contact with the Northern Hemisphere surface.

### b. Transit-time pdf’s to first and since last surface contact: Interior source, zero-mixing ratio BC

Given a constant source in the interior of the geophysical reservoir, with *χ*0BC over some control surface, Ω, tracer age can be defined at any point **r**, but does not everywhere reduce to a source-to-**r** transit time. Fluid particles are labeled with clocks at the source and these clocks are removed at Ω. A fluid particle is not assigned a new clock until it recirculates to the source. Thus, unlabeled fluid always mixes with labeled fluid and *χ* does not continually grow in time, so that *χ* in Eq. (31) for *A* never vanishes. In terms of the flux picture of transit-time distributions, the flux from an interior pulse is only a transit-time pdf (the pdf to first contact) when it is the total flux into Ω. With this configuration it is, therefore, only possible to deduce the source-to-surface transit-time pdf.

Geophysical realizations of interior sources with approximate zero-mixing ratio BCs are provided by stratospheric emissions that rapidly rain out of the atmosphere upon reaching the lower troposphere. For example, the flux of radiocarbon out of the atmosphere in response to the stratospheric nuclear bomb tests of the 1950s contains information on the source-to-surface transit-time pdf. Aerosols from volcanic eruptions and long-lived emissions from high-flying aircraft may also provide such information, if their signals can be sufficiently separated from the background. For the oceans, one could imagine a man-made artificial release of some suitable tracer, with the atmosphere providing a uniform BC on the oceanic mixed layer concentration. The normalized net flux into the atmosphere as a function of time would be equivalent to the release-point-to-ocean-surface transit-time pdf.

Interior sources can easily be incorporated into numerical models of geophysical flows to extract transit times of interest. We demonstrate this by using the GISS CTM to explicitly generate *G*_{0} for the source point **r** = (3.9°N, 175°W, 34 mb) in the lower tropical stratosphere with a *χ*0BC over the earth’s surface (lowest model level). Spatially the source occupies a single CTM grid box (in the vertical from 46 to 22 mb), and in time the source is “on” for an entire month. Thus, the response to this source is a smoothed *G*_{0} resulting from a convolution with a 1-month square pulse in time. We generate 12 such smoothed *G*_{0}, one for the source being on for each month of the year. The net flux into the earth’s surface was computed from the tracer mass loss rate as −∂_{t}*M*_{0}(*t*|**r**′, *t*′) [cf. (25)], which gives us the pdf, ^{†} of transit times to *first* contact with the surface. To contrast this against the pdf of transit times since *last* contact with the surface, we computed a smoothed *χ*0BC over the lowest model level. This was again done for 12 such month-long pulses in the surface BC, one for each month of the year. Monthly means of the resulting pdf’s, **r**_{A}, *t*|Ω, *t*′) and ^{†}(Ω, *t*|**r**_{A}, *t*′), are shown in Fig. 10 as a function of elapsed time *ξ* = *t* − *t*′, averaged over the 12 source times.

The pdf’s ^{†} are quite different, as may be expected. Air travels from the surface to the lower tropical stratosphere via efficient convective mixing in the tropical troposphere and vertical advection through the tropical lower stratosphere. The dominant paths from the surface to **r** through the tropical tropopause are, therefore, expected to have relatively short transit times compared to the paths leading from **r** to the surface, which involve slow stratospheric circulation to the midlatitude tropopause where tracer then mixes with the troposphere again. Correspondingly, the pdf, ^{†}, of times to first surface contact is much broader (indicating a greater multiplicity of pathways) with a peak at 1.3 yr and a mean-transit time of 2.7 yr.

### c. Nonstationary tracer age: An oceanic example

In many geophysical examples mean transit times cannot be readily inferred from tracers because the tracer distribution is in a transient state. In such cases, not all fluid particles carry a tracer clock, so that observations in general represent averages over labeled and unlabeled fluid. Consider anthropogenic tracers, such as the CFCs, in the ocean. The atmospheric evolution of CFCs acts as a time-dependent BC at the ocean surface, which then propagates into the deep ocean. Most ocean water, however, has not made contact with the surface during the time CFCs have been present in the atmosphere, approximately the past 40 yr. Thus, there is “clean” ocean water available to mix with CFC-labeled water, preventing the interpretation of a CFC lag time as a mean transit time since last surface contact.

*S*

_{2}(

**r**,

*t*) =

*γtS*

_{1}(

**r**,

*t*), where

*γ*is a constant rate, and using the definition (20) of mean tracer age, it is straightforward to derive that the tracer ratio,

*R*≡

*χ*

_{2}/

*χ*

_{1}, is related to the tracer age,

*A*

_{1}, of

*χ*

_{1}through

*R*

*γ*

*t*

*A*

_{1}

*t*= 0. Thus,

*R*contains information on the mean transit time of

*χ*

_{1}-labeled fluid elements, but

*R*will

*not*be related to the flow’s mean transit time since last surface contact until a statistical stationary response to the sources is achieved and the diffusive coupling term,

*A*

_{1}. This may take millennia in the deep ocean. How the stationary-sate mean tracer age,

*A*

_{1}, is related to the moments of the transit-time pdf depends on the structure of

*S*

_{1}. (The case of constant

*S*

_{1}is discussed in section 5a.)

## 6. Summary and conclusions

To make the best use of the transport information provided by geophysical tracer observations and to interpret transport properties of numerical models, one needs a conceptual and analytical framework allowing clear physical interpretation. In this paper we have presented such a framework by defining the tracer-age distribution and exploring its relationship with transit-time pdf’s using the passive-tracer Green function. The pdf of transit times *G*′, of arbitrary mixing-ratio BCs on Ω. By expressing *G*′ in terms of the appropriate Green function, we found that **r**, *t*|Ω, *t*′) is the time series of flux into Ω resulting from a unit-mass injection at (**r**, *t*) into the time-reversed flow.

The Green function of the tracer transport equation allows one to decompose tracer mixing ratio into contributions injected into the flow at each time interval in the past. Tracer particles can thus be thought of as clocks that are started when they are injected into the flow. This simple idea led us to a natural and literal definition of a tracer-age distribution, *Z,* and its mean, the mean tracer age, *A,* which are generally distinct from a transit-time pdf and a mean transit time for the flow. The tracer-age distribution, *Z,* bins the mass of clocks according to the time they read. We used *Z* to explicitly construct a pdf of transit times from some point **r** to a surface, Ω. With a constant source at **r** and a carefully chosen volume (surrounding Ω), for which mass fractions of a given clock time interval are computed, *Z* can be reduced to **r**, *t*|Ω, *t*′) as the pdf of times since fluid at **r** had *last contact* with Ω and also establishes its adjoint, ^{†}(Ω, *t*′|**r**, *t*), as the pdf of times for fluid at **r** to have *first contact* with Ω. Except for purely diffusive flow, ^{†} are generally not equivalent.

Using a the GISS CTM, we gave explicit examples of the pdf **r** in the lower tropical stratosphere had last surface contact and the pdf ^{†} for air at **r** to have first surface contact. The two pdf’s are markedly different. Transport to **r** from the surface is dominated by rapid tropical convection to the tropopause, followed by direct upward advection in the lower tropical stratosphere, while subsequent transport back to the troposphere involves the slower stratospheric circulation to reach the midlatitude tropopause. In this example ^{†} was obtained as the flux of tracer into the earth’s surface resulting from a unit-mass injection at (**r**, *t*′) with zero-mixing-ratio BCs over the earth’s surface. If ^{†} were of interest for all **r** in the stratosphere (e.g., to obtain stratospheric residence times for tracer emitted at different locations), it would be impractical to simulate unit-mass injections from every point. However, because of the equivalence of ^{†} and a boundary propagator in the time-reversed flow, ^{†} could be computed at every point as a response to a pulse BC for the adjoint numerical model.

It is natural to consider point-to-point transit times. However, while the point-to-point transit-time pdf is well defined, its moments, including the mean transit time, are infinite in two and three dimensions. For a finite-sized reservoir such as the atmosphere, this infinite mean transit time does not imply that a unit mass released at **r**′ will produce a finite mixing ratio at **r** only after an infinite time. The majority of tracer particles arrive at **r** after a finite time, but there are a sufficient number of particles taking arbitrarily long to find their point target of Ω to make the mean time for particles to have visited **r** via all possible paths infinite. This highlights the fact that any transit-time pdf not only depends on the flow but also on the geometry of the control surface, Ω, and the dimensionality of the problem. We note, however, that when Ω is a large, extended region, the transit times since last, and to first, Ω contact are not very sensitive to the geometry of Ω. For example, whether Ω is the entire earth’s surface or just the Northern Hemisphere has only a minor impact on the mean transit time since last contact for stratospheric air parcels. Even though the magnitude of mean transit times sharply increases when Ω is shrunk to a small surface patch, we find that the *gradients* of the mean transit time are only affected in the vicinity of Ω. For example, stratospheric transit-time differences with respect to the tropical tropopause are insensitive to the size of a surface Ω.

To illustrate how transit-time information can be extracted from geophysical tracers, we compared the equations of motion for mean tracer age and mean transit time. In source-free regions, the equation for mean tracer age has an extra diffusive coupling,

The distribution of anthropogenic tracers in the ocean is in a transient state, which prevents “snapshots” of tracer lag times from being interpreted as mean transit times. Consequently, ratios of tracers, such as CFC-11/CFC-12, are often resorted to for the extraction of transport timescales. Using the framework developed here, we have shown that for the special case, where the ratio of the sources increases linearly in time, the ratio of mixing ratios at a point **r** in the ocean has a concrete interpretation in terms of the mean tracer age since last surface contact. This tracer age has no simple relation to the moments of the transit-time pdf, because not all paths available to water from the surface to **r** have been sampled by tracer. Nonetheless, mean tracer age may be useful to help interpret constraints on ocean models.

We have given precise definitions and interpretations of transit-time and tracer-age distributions, and hope that the framework developed will be helpful in future investigations of geophysical transport. We have presented a few examples, but many issues remain to be explored. These include how best to deal with spatially extended sources, the detailed dependence of mean transit times on the size of the source region, the physics that sets the length scale over which a point source sustains high gradients in the presence of turbulent flow, and generally an investigation of how different paths corresponding to different dynamical mechanisms contribute to the transit-time pdf. In addition to being useful in the context of the atmosphere, the framework of this paper may also help to interpret oceanic transport timescales inferred from tracers and tracer ratios having sources with various time variations. For example, natural radiocarbon, which has known atmospheric abundance due to cosmogenic generation and decays in the ocean with its isotopic half-life of 5568 yr, has been used to establish oceanic timescales (e.g., Broeker et al. 1988) that likely approximate mean transit times closely.

## Acknowledgments

We thank Greg Flato and Ron Miller for enjoyable discussions on subtle aspects of diffusive transport. Tim Hall acknowledges support from the NASA Atmospheric Effects of Aviation Program.

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## APPENDIX A

### General Solution and Advective-Diffusive Case

*G,*we also need the equation of motion for the adjoint Green function,

*G*

^{†}(

**r**,

*t*|

**r**

_{0},

*t*

_{0}), which takes tracer from (

**r**

_{0},

*t*

_{0}) to (

**r**,

*t*) in the time-reversed flow. The equation for

*G*

^{†}is obtained by replacing

^{†}and reversing time,

*t*→ −

*t*[see, e.g., Morse and Feshbach (1953), hereafter MF53)], which gives

*ρ*(

**r**,

*t*)

*G*

^{†}(

**r**,

*t*|

**r**′,

*t*′) and (A1) by

*ρ*(

**r**,

*t*)

*χ*(

**r**,

*t*) and subtract the two resulting equations to obtain

**,**

**∇**^{†}all act on

**r**. Equation (A3) may be considered a generalization of Green’s theorem. Now integrate (A2) with respect to

**r**over the entire domain and with respect to

*t*from

*t*= 0 to

*t*=

*t*

^{+}

_{0}

*t*

^{+}

_{0}

*t*

_{0}. Under this

*dt*

*d*

^{3}

*r*integral, the last term of (A2) becomes

*χ*(

**r**

_{0},

*t*

_{0}), which is then equal to

*G*

^{†}(

**r**,

*t*

^{+}

_{0}|

**r**

_{0},

*t*

_{0})

**·**

**∇****C**with the surface integral of

**n̂**·

**C**over the boundary of the tracer domain (denoted by ∂). Now relabel variables (

**r**

_{0},

*t*

_{0}) → (

**r**,

*t*) and (

**r**,

*t*) → (

**r**′,

*t*′) and express

*G*

^{†}in terms of

*G*by using the reciprocity relation

*G*

^{†}(

**r**

_{1},

*t*

_{1}|

**r**

_{2},

*t*

_{2}) =

*G*(

**r**

_{2},

*t*

_{2}|

**r**

_{1},

*t*

_{1}) to obtain (4) with

**C**defined through (A3). (A reciprocity relation of this form generally holds for Green functions—for a derivation see, e.g., MF53.)

**v**with −

**v**, while the diffusive operator remains unchanged. Substituting (3) and its adjoint into (A3), we obtain

**∇****C**

*ψ, ϕ*

**∇***κρ*

*ψ*

**∇***ϕ*

*ϕ*

**∇***ψ*

*ρ*

**v**

*ψϕ*

_{t}

*ρ*+

**(**

**∇****v**

*ρ*) = 0. Using the form (A5) for

**·**

**∇****C**in (A3) gives the general advection-diffusion boundary term (5).

## APPENDIX B

### Analytical Solutions for a Simple 1D Model

_{t}

*χ*

*ρ*

^{−1}

*κ*

_{z}

*ρ*

_{z}

*χ*

*S,*

*z*≥ 0; the air density,

*ρ,*has the form

*ρ*(

*z*) =

*ρ*

_{0}exp(−

*z*/

*H*); and the diffusivity

*κ,*scale height

*H,*and density scale

*ρ*

_{0}are constants. We nondimensionalize (B1) via

*z*/

*H*→

*z, t*(

*κ*/

*H*

^{2}) →

*t, ρ*/

*ρ*

_{0}→

*ρ.*Since the transport operator here is constant in time,

*G, G*

_{0}, and

*G*′ are simple functions that can depend on time only through

*ξ*=

*t*−

*t*′. The Green functions of (B1) are defined by replacing the source

*S*with

*δ*(

*ξ*)

*δ*

^{3}(

**r**−

**r**′)/

*ρ*(

**r**′) which in 1D reduces to

*δ*(

*ξ*)

*δ*(

*z*−

*z*′)/

*ρ*(

*z*′). Using standard methods, we calculate

*G*as

*G*

_{0}, with control “surface” at

*z*= 0, as

*G*and

*G*

_{0}are nondimensionalized by multiplication with

*Hρ*

_{0}.) Note that the long-time limits

*G*

_{∞}= 1 and lim

_{ξ→∞}

*G*

_{0}= 0.

*G*′ and

*G*

_{0}becomes

*G*

*z, ξ*

_{z′}

*G*

_{0}

*z, z*

*ξ*

_{z′=0}

*G*′, which in 1D is also the transit time pdf

*G*′ is nondimensionalized through (

*H*

^{2}/

*κ*)

*G*′ →

*G*′]. We may rewrite (B4) as

*G*′(

*z, ξ*) = lim

_{z′→0}

*G*

_{0}(

*z, z*′,

*ξ*)/

*z*′, which is the 1D-model version of (15).

*M*=

^{∞}

_{0}

*dz*

*e*

^{−z}

*G*(

*z, z*′,

*ξ*) = 1, while

*M*

_{0}=

^{∞}

_{0}

*dz*

*e*

^{−z}

*G*

_{0}(

*z, z*′,

*ξ*) is given by

*z, ξ*) = −∂

_{ξ}

*M*

^{†}

_{0}(

*z,*−

*ξ*)

*P*(

*ξ*), is given by

*M*

_{A}= 1, here. Note the singularity

*P*(

*ξ*) → 1/

*πξ*

*ξ*→ 0

^{+}, but that

^{∞}

_{0}

*dξ*

*P*(

*ξ*) = 1 and 〈

*ξ*〉 ≡

^{∞}

_{0}

*dξ*

*ξP*(

*ξ*) = 1. Thus, the most probable “particle age” of a surface-marked particle is

*ξ*= 0, while its expected particle age is 〈

*ξ*〉 = 1.

## APPENDIX C

### Transit-Time Pdf and Mean Transit Times in Two and Three Dimensions

#### Unbounded domain with radial symmetry in three dimensions

*r*=

*a*centered at the origin of a spherical coordinate system. For simplicity we assume spherical symmetry, so that

*G*

_{0}is the response to a unit-mass injection distributed over a shell of radius

*r*=

*r*′. The nondimensionalized diffusion equation for

*G*

_{0}can then be written as

*G*

_{0}(

*a, r*′,

*t*−

*t*′) = 0, which has solution

*r, ξ*) = −∂

_{ξ}

^{∞}

_{a}

*dr*′ 4

*πr*′

^{2}

*G*

_{0}(

*r, r*′,

*ξ*). This quantity has the interpretation of the flux from a unit injection over a shell at

*r*into the bubble at

*r*=

*a.*The total amount of mass entering the bubble is

*r*= ∞. Since the time-normalized flux into the bubble still has the natural interpretation of an arrival-time pdf (for pure diffusion, the adjoint problem with time running backward is the same as the direct problem with time running forward), the appropriate transit-time pdf for this problem is not

*r, t*) but (

*r*/

*a*)

*r, t*). The limit lim

_{a→0}(

*r*/

*a*)

*r, t*) is well defined, even though the fraction of tracer mass reaching the infinitesimal bubble goes to zero. Note, however, that independently of the value of

*a,*the mean transit time, Γ ≡

^{∞}

_{0}

*dξ*

*ξ*

*r, ξ*), is infinite in this setting, due to the fact that particles can take arbitrarily long paths away from the boundary to give

#### Mean transit times for a bounded domain with radial symmetry in two and three dimensions

*κ,*and constant density in a domain of radial symmetry bounded by inner and outer radii, so that

*a*⩽

*r*⩽

*b.*Over the surface,

*r*=

*a,*a

*δ*-function BC is applied on the mixing ratio to obtain

*r*=

*b*a zero-flux BC is enforced. In both two and three dimensions (2D and 3D), it is straightforward to obtain the Laplace transform of

*r, t*) are easily obtained from its Laplace transform,

*r, s*), since for integer

*n,*we have

^{∞}

_{0}

*dξ*

*ξ*

^{n}

*r, ξ*) = (−1)

^{n}lim

_{s→0}∂

^{n}

*r, s*)/∂

*s*

^{n}. For the bounded domain the normalization

^{∞}

_{0}

*dξ*

*r, ξ*) = 1 is confirmed and the mean transit time, Γ, is given in 3D by

*b,*Γ is finite for finite

*a,*but diverges like 1/

*a*in 3D and like log(

*a*) in 2D. Similarly, the

*n*th moments

^{∞}

_{0}

*dξ*

*ξ*

^{n}

*r, ξ*) and

^{∞}

_{0}

*dξ*(

*ξ*− Γ)

^{n}

*r, ξ*) diverge like 1/

*a*

^{n}in 3D and like [log(

*a*)]

^{n}in 2D.

*a*→ 0 is afforded by the correspondence (35) discussed in section 5 between the differential tracer age resulting from a constant source, Δ

*A*∝

*χ*

^{+}, and Γ. Consider, for simplicity, the case of advection in the presence of isotropic diffusion (

*κ*), when the time-averaged equation for

*χ*

^{+}=

*χ*−

*χ*

_{0}may be written as

_{t}

*χ*

_{0}is the spatially uniform growth rate of the background mixing ratio, and the overbar denotes time average. On sufficiently small spatial scales the diffusion term (highest-order derivative) dominates over advection, so that in the immediate vicinity of a constant point source,

*S*=

*s*

_{0}

*δ*(

**r**−

**r**

_{0}), mixing ratio is determined by ∇

^{2}

*χ*

^{+}

*δ*(

**r**−

**r**

_{0}). This means that as the patch radius,

*a*→ 0,

*χ*

^{+}, and hence Δ

*A,*diverge in three dimensions like 1/

*a,*and in two dimensions like log(

*a*). These are precisely the limiting dependencies of Γ calculated above for the specific purely diffusive models. (An example of a more general two-dimensional case is a zonally averaged model with a point source, corresponding to a line source along a circle of constant latitude.)

Analytical solutions of *G*_{0}(*z, z*′, *ξ*) for the stationary 1D model atmosphere of appendix B, as a function of elapsed time, *ξ.* The source point is located at *z*′ = 3.0. The zero-mixing-ratio BC is applied at the surface, *z* = 0. The time evolution of *G*_{0} is plotted at the three locations, *z,* indicated

Citation: Journal of the Atmospheric Sciences 57, 21; 10.1175/1520-0469(2000)057<3539:TTATAD>2.0.CO;2

Analytical solutions of *G*_{0}(*z, z*′, *ξ*) for the stationary 1D model atmosphere of appendix B, as a function of elapsed time, *ξ.* The source point is located at *z*′ = 3.0. The zero-mixing-ratio BC is applied at the surface, *z* = 0. The time evolution of *G*_{0} is plotted at the three locations, *z,* indicated

Citation: Journal of the Atmospheric Sciences 57, 21; 10.1175/1520-0469(2000)057<3539:TTATAD>2.0.CO;2

Analytical solutions of *G*_{0}(*z, z*′, *ξ*) for the stationary 1D model atmosphere of appendix B, as a function of elapsed time, *ξ.* The source point is located at *z*′ = 3.0. The zero-mixing-ratio BC is applied at the surface, *z* = 0. The time evolution of *G*_{0} is plotted at the three locations, *z,* indicated

Citation: Journal of the Atmospheric Sciences 57, 21; 10.1175/1520-0469(2000)057<3539:TTATAD>2.0.CO;2

Analytical solutions for the boundary propagator *z, ξ*) for the stationary 1D model atmosphere of appendix B, as a function of elapsed time, *ξ.* The zero-mixing-ratio BC is applied at the surface, *z* = 0. The time evolution of *z,* indicated

Analytical solutions for the boundary propagator *z, ξ*) for the stationary 1D model atmosphere of appendix B, as a function of elapsed time, *ξ.* The zero-mixing-ratio BC is applied at the surface, *z* = 0. The time evolution of *z,* indicated

Analytical solutions for the boundary propagator *z, ξ*) for the stationary 1D model atmosphere of appendix B, as a function of elapsed time, *ξ.* The zero-mixing-ratio BC is applied at the surface, *z* = 0. The time evolution of *z,* indicated

The probability *M*_{0}(*z*′, *ξ*) in the analytical 1D atmosphere of appendix B, that a marked fluid particle located at *z*′ at time *ξ* = 0 will not have made contact with the surface, *z* = 0, a time *ξ* later. As expected, the farther *z*′ is from the surface, the longer it takes until the particle has significant probability of surface contact

The probability *M*_{0}(*z*′, *ξ*) in the analytical 1D atmosphere of appendix B, that a marked fluid particle located at *z*′ at time *ξ* = 0 will not have made contact with the surface, *z* = 0, a time *ξ* later. As expected, the farther *z*′ is from the surface, the longer it takes until the particle has significant probability of surface contact

The probability *M*_{0}(*z*′, *ξ*) in the analytical 1D atmosphere of appendix B, that a marked fluid particle located at *z*′ at time *ξ* = 0 will not have made contact with the surface, *z* = 0, a time *ξ* later. As expected, the farther *z*′ is from the surface, the longer it takes until the particle has significant probability of surface contact

The particle-age pdf, *P*(*ξ*), of the 1D atmosphere of appendix B. Note the *ξ*^{−1/2} divergence as *ξ* → 0. (The dotted line indicates the asymptotic *ξ*^{−1/2} power law.) This divergence is a generic feature of transport with a diffusive component and not a peculiarity of the 1D model (see section 2d)

The particle-age pdf, *P*(*ξ*), of the 1D atmosphere of appendix B. Note the *ξ*^{−1/2} divergence as *ξ* → 0. (The dotted line indicates the asymptotic *ξ*^{−1/2} power law.) This divergence is a generic feature of transport with a diffusive component and not a peculiarity of the 1D model (see section 2d)

The particle-age pdf, *P*(*ξ*), of the 1D atmosphere of appendix B. Note the *ξ*^{−1/2} divergence as *ξ* → 0. (The dotted line indicates the asymptotic *ξ*^{−1/2} power law.) This divergence is a generic feature of transport with a diffusive component and not a peculiarity of the 1D model (see section 2d)

Schematic illustration of two equivalent ways in which one can obtain the pdf, *ξ,* since fluid at (**r**, *t*) was last in contact with some arbitrary fixed surface, Ω, indicated by the shaded region. (top) The time-forward flow. If a tracer mixing ratio on Ω is specified to be an impulse at *t*′ proportional to *δ*(*t* − *t*′), then the time evolution of the resulting mixing ratio at **r** as a function of elapsed time, *ξ* = *t* − *t*′, is proportional to **r**, *t*′ + *ξ*|Ω, *t*′). (bottom) The equivalent situation in the time-reversed flow where zero-mixing-ratio BCs are imposed on Ω. A unit mass injected at (**r**, *t*) results in a net flux into Ω at a time *ξ* = *t* − *t*′ earlier (since time runs backward). This net flux is proportional to **r**, *t*|Ω, *t* − *ξ*) = **r**, *t*′ + *ξ*|Ω, *t*′)

Schematic illustration of two equivalent ways in which one can obtain the pdf, *ξ,* since fluid at (**r**, *t*) was last in contact with some arbitrary fixed surface, Ω, indicated by the shaded region. (top) The time-forward flow. If a tracer mixing ratio on Ω is specified to be an impulse at *t*′ proportional to *δ*(*t* − *t*′), then the time evolution of the resulting mixing ratio at **r** as a function of elapsed time, *ξ* = *t* − *t*′, is proportional to **r**, *t*′ + *ξ*|Ω, *t*′). (bottom) The equivalent situation in the time-reversed flow where zero-mixing-ratio BCs are imposed on Ω. A unit mass injected at (**r**, *t*) results in a net flux into Ω at a time *ξ* = *t* − *t*′ earlier (since time runs backward). This net flux is proportional to **r**, *t*|Ω, *t* − *ξ*) = **r**, *t*′ + *ξ*|Ω, *t*′)

Schematic illustration of two equivalent ways in which one can obtain the pdf, *ξ,* since fluid at (**r**, *t*) was last in contact with some arbitrary fixed surface, Ω, indicated by the shaded region. (top) The time-forward flow. If a tracer mixing ratio on Ω is specified to be an impulse at *t*′ proportional to *δ*(*t* − *t*′), then the time evolution of the resulting mixing ratio at **r** as a function of elapsed time, *ξ* = *t* − *t*′, is proportional to **r**, *t*′ + *ξ*|Ω, *t*′). (bottom) The equivalent situation in the time-reversed flow where zero-mixing-ratio BCs are imposed on Ω. A unit mass injected at (**r**, *t*) results in a net flux into Ω at a time *ξ* = *t* − *t*′ earlier (since time runs backward). This net flux is proportional to **r**, *t*|Ω, *t* − *ξ*) = **r**, *t*′ + *ξ*|Ω, *t*′)

Schematic of the construct to demonstrate the equivalence between the tracer-age distribution, *Z,* computed for the volume, *V,* and the flux into the control surface, Ω. As discussed in section 4, we take the limit as the thickness of *V* (chosen to be proportional to the diffusivity, *κ*) goes to zero, followed by a collapse of the resulting thin shell onto Ω as *ϵ* → 0

Schematic of the construct to demonstrate the equivalence between the tracer-age distribution, *Z,* computed for the volume, *V,* and the flux into the control surface, Ω. As discussed in section 4, we take the limit as the thickness of *V* (chosen to be proportional to the diffusivity, *κ*) goes to zero, followed by a collapse of the resulting thin shell onto Ω as *ϵ* → 0

Schematic of the construct to demonstrate the equivalence between the tracer-age distribution, *Z,* computed for the volume, *V,* and the flux into the control surface, Ω. As discussed in section 4, we take the limit as the thickness of *V* (chosen to be proportional to the diffusivity, *κ*) goes to zero, followed by a collapse of the resulting thin shell onto Ω as *ϵ* → 0

Annual averages of the differential tracer age, 2Δ*A,* and the mean transit time, Γ, as simulated by the CCC GCM for the case of the control surface, Ω, being a small patch in Europe centered on (50.1°N, 11.3°W) with 0.28% of the global surface area. The tracer age was computed as 2Δ*A*(**r**, *t*) = [*χ*(Ω, *t*) − *χ*(**r**, *t*)]/(*s*_{0}*G*_{∞}) from the stationary-state mixing ratio, *χ,* resulting from a constant source over Ω. [The reference value, *χ*(Ω, *t*), is the average of *χ* over Ω.] The transit time, Γ, was computed as [*χ*(Ω, *t*) − *χ*(**r**, *t*)]/*γ* from the stationary-state mixing ratio resulting from the ramp BC, *χ*(**r**_{s}, *t*) = *γt* for **r**_{s} on Ω. The contour interval is 0.25 yr

Annual averages of the differential tracer age, 2Δ*A,* and the mean transit time, Γ, as simulated by the CCC GCM for the case of the control surface, Ω, being a small patch in Europe centered on (50.1°N, 11.3°W) with 0.28% of the global surface area. The tracer age was computed as 2Δ*A*(**r**, *t*) = [*χ*(Ω, *t*) − *χ*(**r**, *t*)]/(*s*_{0}*G*_{∞}) from the stationary-state mixing ratio, *χ,* resulting from a constant source over Ω. [The reference value, *χ*(Ω, *t*), is the average of *χ* over Ω.] The transit time, Γ, was computed as [*χ*(Ω, *t*) − *χ*(**r**, *t*)]/*γ* from the stationary-state mixing ratio resulting from the ramp BC, *χ*(**r**_{s}, *t*) = *γt* for **r**_{s} on Ω. The contour interval is 0.25 yr

Annual averages of the differential tracer age, 2Δ*A,* and the mean transit time, Γ, as simulated by the CCC GCM for the case of the control surface, Ω, being a small patch in Europe centered on (50.1°N, 11.3°W) with 0.28% of the global surface area. The tracer age was computed as 2Δ*A*(**r**, *t*) = [*χ*(Ω, *t*) − *χ*(**r**, *t*)]/(*s*_{0}*G*_{∞}) from the stationary-state mixing ratio, *χ,* resulting from a constant source over Ω. [The reference value, *χ*(Ω, *t*), is the average of *χ* over Ω.] The transit time, Γ, was computed as [*χ*(Ω, *t*) − *χ*(**r**, *t*)]/*γ* from the stationary-state mixing ratio resulting from the ramp BC, *χ*(**r**_{s}, *t*) = *γt* for **r**_{s} on Ω. The contour interval is 0.25 yr

As for Fig. 8 with Ω consisting of the Northern Hemisphere landmass except that of Africa and South America. (Note that the grayscale is different from that of Fig. 8.)

As for Fig. 8 with Ω consisting of the Northern Hemisphere landmass except that of Africa and South America. (Note that the grayscale is different from that of Fig. 8.)

As for Fig. 8 with Ω consisting of the Northern Hemisphere landmass except that of Africa and South America. (Note that the grayscale is different from that of Fig. 8.)

The pdf of transit times, ^{†} (solid line), to first contact with the earth’s surface from the point **r** = (3.9°N, 175°W, 34 mb) in the tropical lower stratosphere, and the pdf of transit times, **r** since last contact with the earth surface, as simulated by the GISS CTM. The pdf ^{†} pdf’s were computed with source and BC pulses for each month of the year. Shown is the dependence on elapsed time *ξ* = *t* − *t*′ averaged over the 12 source times. Arrows indicate a mean transit time of 0.74 yr for ^{†}. The inset shows a schematic of zonally averaged atmospheric transport. The gray shading indicates the tropopause. The dashed arrows represent paths that contribute to **r** back to the surface via the stratospheric circulation and stratosphere–troposphere exchange. The return paths contribute to ^{†}

The pdf of transit times, ^{†} (solid line), to first contact with the earth’s surface from the point **r** = (3.9°N, 175°W, 34 mb) in the tropical lower stratosphere, and the pdf of transit times, **r** since last contact with the earth surface, as simulated by the GISS CTM. The pdf ^{†} pdf’s were computed with source and BC pulses for each month of the year. Shown is the dependence on elapsed time *ξ* = *t* − *t*′ averaged over the 12 source times. Arrows indicate a mean transit time of 0.74 yr for ^{†}. The inset shows a schematic of zonally averaged atmospheric transport. The gray shading indicates the tropopause. The dashed arrows represent paths that contribute to **r** back to the surface via the stratospheric circulation and stratosphere–troposphere exchange. The return paths contribute to ^{†}

The pdf of transit times, ^{†} (solid line), to first contact with the earth’s surface from the point **r** = (3.9°N, 175°W, 34 mb) in the tropical lower stratosphere, and the pdf of transit times, **r** since last contact with the earth surface, as simulated by the GISS CTM. The pdf ^{†} pdf’s were computed with source and BC pulses for each month of the year. Shown is the dependence on elapsed time *ξ* = *t* − *t*′ averaged over the 12 source times. Arrows indicate a mean transit time of 0.74 yr for ^{†}. The inset shows a schematic of zonally averaged atmospheric transport. The gray shading indicates the tropopause. The dashed arrows represent paths that contribute to **r** back to the surface via the stratospheric circulation and stratosphere–troposphere exchange. The return paths contribute to ^{†}