## 1. Introduction

Natural and anthropogenic tracers have been used to estimate the ventilation history of ocean water masses. The “ages” constructed from tracers generally reveal different and complementary information about the ventilation. Recent work has made explicit the fact that a water mass must be characterized by a distribution of times since since it last made surface contact, rather than a single “age” (Beining and Roether 1996; Delhez et al. 1999; Khatiwala et al. 2001; Haine and Hall 2002), and observable ages represent differently weighted averages over this distribution (Haine and Hall 2002). Many of these ideas build on previous work intepreting stratospheric tracers (Hall and Plumb 1994) and recent more general work on transport timescales in geophysical flows (Holzer and Hall 2000; Beckers et al. 2001; Deleersnijder et al. 2001a).

*τ*

_{id}(

*r,*

*t*) (e.g., England 1995), which we refer to simply as the “ideal age.” It is defined by

*t*) is the unit step function [Θ(

*t*) = 1 for

*t*≥ 0 and Θ(

*t*) = 0 for

*t*< 0]. The boundary condition (BC) is

*τ*

_{id}(

*S,*

*t*) = 0, where

*S*is the ocean surface, and the initial condition is

*τ*

_{id}(

*r,*0) = 0. In this note we restrict attention to the case of stationary transport (i.e., the coefficients of

In the limit of long elapsed time compared to timescales of the circulation (“steady state”), *τ*_{id} is a natural diagnostic. The irreducible fluid elements that compose the water mass have had their “clocks” increased one time unit per unit time by the source [rhs of (1)] since last boundary contact, where their clocks were reset to zero. Thus, *τ*_{id} averaged over the elements of the water mass [the “observable” quantity, were there such a tracer obeying (1)] is the “ideal age,” the average time since the water mass last made surface contact. Due to mixing, the clock times of the individual elements comprising the water mass may vary widely.

The ideal age is a popular and useful diagnostic in ocean models, but generally only the steady-state solution is reported and analyzed. However, the transient approach to steady state contains the transport information provided by the steady-state solution and in principle much more. This transient is a useful diagnostic if it can be interpreted physically in a straightforward manner. Recently, mathematical and physical frameworks have been developed in which passive tracer fields can be expressed in terms of transient evolutions that have interpretations as transit time or age distributions (Holzer and Hall 2000; Beckers et al. 2001; Deleersnijder et al. 2001a). In this note we present an example of this connection between a tracer field and an age-related transient that should be of practical interest to ocean modelers. Namely, we derive a simple and direct relationship between *τ*_{id} and the distribution of transit times since a water mass made last surface contact. This distribution has been termed the “age spectrum” in stratosphere applications (Kida 1983; Hall and Plumb 1994), and we use that nomenclature here.

## 2. Idealized age tracer and age spectrum

*τ*

_{id}in terms of Green's functions. Given the unit uniform source of (1) one has

*D*is the physical domain (i.e., the ocean),

*ρ*the fluid density, and

*G*(

*r,*

*t,*

*r*′,

*t*′) the Green's function, the response at (

*r,*

*t*) to a point source at (

*r*′,

*t*′). The function

*G*obeys a BC compatible with

*τ*

_{id}; namely,

*G*(

*r*

_{S},

*t,*

*r*′,

*t*′) = 0 for

*r*

_{S}on

*S.*For stationary transport

*G*depends only on elapsed time

*ξ*=

*t*−

*t*′, and (2) can be rewritten

*r*to a spatially distributed source. We replace this with the response integrated over the domain to a point source at

*r*by exploiting the reciprocity condition for Green's functions:

*G*(

*r,*

*r*′,

*ξ*) =

*G*

^{†}(

*r*′,

*r,*−

*ξ*), where

*G*

^{†}is the Green's function for the adjoint flow (e.g., Morse and Feshbach 1953). The reciprocity condition says that the response at

*r*to a point source at

*r*′ in the time-forward flow is the same as the response at

*r*′ to a source at

*r*in the time-reversed adjoint flow (TRAF). Thus,

*M*

^{†}≡ ∫

_{D}

*d*

^{3}

*rρG*

^{†}is the total tracer mass in

*D*at elapsed time −

*ξ*“after” the unit tracer release from

*r*in the TRAF.

*M*

^{†}(

*r,*−

*ξ*) as

*ξ*increases (−

*ξ*decreases). Initially in the TRAF the unit tracer mass is localized near the release point

*r,*and

*M*

^{†}= 1. Subsequently, tracer begins to make contact with the boundary where it is lost, and

*M*

^{†}declines from unity. The rate of change ∂

_{ξ}

*M*

^{†}(

*r,*−

*ξ*) (subscripts indicate differentiation) must equal the flux of tracer mass out through the boundary. That is,

**n**is the unit normal vector on

*S*directed into

*D,*

*κ*is the diffusivity, and the gradient is evaluated on

*S.*[Note that the flux into

*S*is purely diffusive;

*G*

^{†}(

*r*

_{S},

*r,*−

*ξ*) = 0 by the boundary condition, so that the advective flux

**v**

*G*

^{†}vanishes.] As discussed by Holzer and Hall (2000), a general Green's function solution for a tracer with arbitrary sources and boundary conditions reveals that the rhs of (5) is the kernel in a convolution with the tracer's time-dependent BC on

*S.*The rhs of (5) acts as a propagator,

*r,*

*ξ*), of BCs on

*S*in the time-forward flow. Thus, (5) can be written ∂

_{ξ}

*M*

^{†}(

*r,*−

*ξ*) =

*r,*

*ξ*), or

The final step in out development is to note that, in addition to being a BC propagator, *ξ* since fluid at *r* last made contact with *S* (Hall and Plumb 1994; Holzer and Hall 2000). Indeed, it was the connection of *M*^{†} that led Holzer and Hall (2000) to this interpretation. Physically, *M*^{†}(*r,* −*ξ*) is the fraction of tracer released from *r* in the TRAF that has not made boundary contact in the elapsed time −*ξ.* Therefore, *δξ*∂_{ξ}*M*^{†}(*r,* −*ξ*) is the fraction that makes first boundary contact in the elapsed time interval −*ξ* → −*ξ* − *δξ*; that is, ∂_{ξ}*M*^{†}, which equals

Expression (8), together with the transit time distribution interpretation, is the key result of this work. The transient solution to the ideal age equation and the age spectrum are related in a simple fashion, a result not previously noted, but which follows naturally from the more general Green's function frameworks of Holzer and Hall (2000) and Beckers et al. (2001). The transient solution of the ideal age contains valuable transport information—far more, in fact, than the steady-state ideal age alone—and this information is readily interpretable in terms of the ventilation history of the water mass. Relationship (8) can be physically understood in a straightforward fashion. At early enough times the fluid at an interior point *r* has not yet felt the influence of the boundary, so that the unit source [rhs of Eq. (1)] causes a linear increase in time of *τ*_{id}. From (8), the boundary propagator *τ*_{id} starts to increase more slowly than linear. Eventually, when *τ*_{id} reaches steady state (∂/∂*t* = 0), the boundary has made its full influence felt, and there is no further boundary information to propagate, whence

## 3. Ideal age convergence

*r,*

*t*) ≡

^{t}

_{0}

*dξξ*

*r,*

*ξ*), where Γ(

*r,*

*t*) is the mean transit time since last boundary contact of the tracer that has accumulated at

*r*by time

*t.*As

*t*→ ∞ all fluid elements of the water mass have made boundary contact at some past time, and Γ becomes the mean transit time since the entire water mass was last at the boundary (the “mean age”), equal to the ideal age. Khatiwala et al. (2001) made the numerical observation that although

*τ*

_{id}and Γ tend to the same value,

*τ*

_{id}converges more rapidly. This is readily verified from the general analysis here. Taking the partial first moment (

^{t}

_{0}

*dξξ*) of (8) one finds

*t*∼ 0 and Γ ∼

*τ*

_{id}. However, at intermediate times Γ <

*τ*

_{id}. Therefore, Γ converges more slowly than

*τ*

_{id}. This has practical consequences for diagnostics of numerical models. If it is the sole intention of a modeler to obtain the ideal age (mean age) then the approach of Eq. (1) is more efficient. (However, it should be recognized that the full age spectrum contains far more information than just its mean.)

## 4. Illustration

The expressions relating *τ*_{id},

*u*is the velocity,

*k*the diffusivity, and Θ(

*t*) the unit step function. The BCs are

*τ*

_{id}(0,

*t*) = 0 and that

*τ*

_{id}(

*x,*

*t*) should not grow exponentially with

*x,*and the initial condition is

*τ*

_{id}(

*r,*0) = 0. One method of solution is to take the Laplace transform, consider the sum of homogeneous and particular solutions, and use the BCs to constrain general constants. This yields the Laplace space solution

*τ̃*

_{id}is the Laplace transform of

*τ*

_{id}and

*s*is the transform variable. We were not able to obtain a closed-form solution for the inverse transform. An integral expression of the solution is

*t*) =

*δ*(

*t*) and no exponential growth with

*x.*The solution can be found in several studies and is

*τ*

_{id}we also note that the Laplace space solution to (13) is

*x,*

*s*

*s*

^{2}

*τ̃*

_{id}

*x,*

*s*

The age spectrum for this 1D model is plotted in Fig. 1a, with the mean age indicated by the symbol. Note that the mean age for this idealized model is simply *x*/*u.* Despite its simplicity, the age spectrum for this model is qualitatively similar to the spectrum computed in the Atlantic sector GCM of Khatiwala et al. (2001). Figure 1b shows *τ*_{id}, obtained from relationship (8), and Γ. Note that *τ*_{id} and Γ both converge to the mean age, but *τ*_{id} does so more rapidly.

## 5. Discussion and summary

We have derived a simple relationship, valid for any type of stationary transport, between the transient ideal age *τ*_{id} of ocean water masses and the age spectrum, *r,* *t*) = −∂_{tt}*τ*_{id}(*r,* *t*). This encompasses, but is more general than, the steady-state relationship that the ideal age equals the mean age (the mean over the age spectrum) (e.g., Boering et al. 1996; Khatiwala et al. 2001). The relationship follows naturally from the general frameworks developed by Holzer and Hall (2000), Beckers et al. (2001), and Deleersnijder et al. (2001a) that provide connections between various tracers using the machinary of Green's functions. The relationship derived here implies that the transient ideal age has diagnostic value—much more, in fact, than the equilibrium ideal age itself. No single timescale can completely summarize both the bulk advection and mixing that connect the surface to subsurface regions. By contrast, a tracer's transient solution displays explicitly its sensitivity to the full distribution of timescales, though this information may be convolved with the tracer's particular boundary condition and source distribution. Green's functions, exploited here, are powerful tools to relate tracers to each other and to extract tracer-independent transport information.

The relationships between *τ*_{id}, *t* − ^{t}_{0}*dξ* ^{ξ}_{0}*dξ*′*r,* *ξ*′) and ^{t}_{0}*dξ**r,* *ξ*). One could simulate

Finally, we note that the relationships derived here are strictly valid only for stationary transport. This is clear from relationship (8). Here *τ*_{id} will exhibit “wiggles,” and its first and second time derivatives will at times be negative. Thus, relationship (8) cannot hold. Nonetheless, if the circulation cycles have periods either much longer than or much shorter than the mean age, then (8) may still hold approximately. If the cycle periods are very long, then the circulation appears approximately stationary over the timescales of interest. If the cycle periods are very short (e.g., seasonal), then suitable filtering will result in an approximate relationship.

## Acknowledgments

This work was supported by a grant from the Physical Oceanography Program of the National Science Foundation (OCE-9911598).

## REFERENCES

Abramowitz, M., and I. A. Stegun, 1972:

*Handbook of Mathematical Functions*. Dover, 1046 pp.Beckers, J. M., E. J. M. Delhez, and E. Deleersnijder, 2001: Some properties of generalized age-distribution equations in fluid dynamics.

,*SIAM J. Appl. Math.***61****,**1526–1544.Beining, P., and W. Roether, 1996: Temporal evolution of CFC-11 and CFC-12 concentrations in the ocean interior.

,*J. Geophys. Res.***101****,**16455–16464.Boering, K. A., S. C. Wofsy, B. C. Daube, H. R. Schneider, M. Lowenstein, and J. R. Podolske, 1996: Stratospheric transport rates and mean age distribution derived from observations of atmospheric CO2 and N2O.

,*Science***274****,**1340–1343.Deleersnijder, E., J. M. Campin, and E. J. M. Delhez, 2001a: The concept of age in marine modelling, I. Theory and preliminary model results.

,*J. Mar. Syst.***28****,**229–267.Deleersnijder, E., E. J. M. Delhez, M. Crucifix, and J. M. Beckers, 2001b: On the symmetry of the age field of a passive tracer released into a one-dimensional fluid flow by a point-source.

,*Bull. Soc. Roy. Sci. Liege***70****,**5–21.Delhez, E. J. M., J. M. Campin, A. C. Hirst, and E. Deleersnijder, 1999: Toward a general theory of age in ocean modelling.

,*Ocean Modell.***1****,**17–27.England, M. H., 1995: The age of water and ventilation timescales in a global ocean model.

,*J. Phys. Oceanogr.***25****,**2756–2777.Haine, T. W. N., and T. M. Hall, 2002: A generalized transport theory: Water mass composition and age.

,*J. Phys. Oceanogr.***32****,**1932–1946.Hall, T. M., and R. A. Plumb, 1994: Age as a diagnostic of stratospheric transport.

,*J. Geophys. Res.***99****,**1059–1070.Holzer, M., and T. M. Hall, 2000: Transit-time and tracer-age distributions in geophysical flows.

,*J. Atmos. Sci.***57****,**3539–3558.Khatiwala, S., M. Visbeck, and P. Schlosser, 2001: Age tracers in an ocean GCM.

,*Deep-Sea Res.***48A****,**1423–1441.Kida, H., 1983: General circulation of air parcels and transport characteristics derived from a hemispheric GCM, Part 2: Very long-term motions of air parcels in the troposphere and stratosphere.

,*J. Meteor. Soc. Japan***61****,**510–522.Morse, P. M., and H. Feshbach, 1953:

*Methods of Theoretical Physics, Part I*. McGraw-Hill, 997 pp.