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2250 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 35

Relationships between Tracer Ages and Potential Vorticity in Unsteady Wind-Driven Circulations

HONG ZHANG,THOMAS W. N. HAINE, AND DARRYN W. WAUGH Department of Earth and Planetary Sciences, The Johns Hopkins University, Baltimore, Maryland

(Manuscript received 4 June 2004, in final form 17 May 2005)

ABSTRACT

The relationships between different tracer ages and between tracer age and potential vorticity are examined by simulating barotropic double-gyre circulations. The unsteady model flow crudely represents aspects of the midlatitude, middepth ocean circulation including inhomogeneous and anisotropic variability. Temporal variations range in scale from weeks to years, although the statistics are stationary. These variations have a large impact on the time-averaged tracer age fields. Transport properties of the tracer age fields that have been proved for steady flow are shown to also apply to unsteady flow and are illustrated in this circulation. Variability of tracer ages from ideal age tracer, linear, and exponential transient tracers is highly coordinated in phase and amplitude and is explained using simple theory. These relationships between different tracer ages are of practical benefit to the problem of interpreting tracer ages from the real ocean or from general circulation models. There is also a close link between temporal anomalies of tracer age and potential vorticity throughout a significant fraction of the domain. There are highly significant anticorrelations between ideal age and potential vorticity in the subtropical gyre and midbasin jet region, but correlation in the central subpolar gyre and eastern part of the domain is not significant. The existence of the relationship is insensitive to the character of the flow, the tracer sources, and the potential vorticity dynamics. Its structure may be understood by considering the different time-mean states of the tracer age and potential vorticity, the different tracer sources and sinks, and the effect of variability in the flow. Prediction of the correlation without knowledge of the time-mean fields is a harder problem, however. Detecting the correlation between potential vorticity and tracer age in the real ocean will be difficult with typical synoptic oceanographic transect data that are well-sampled in space, but sparse in time. Neverthe- less, it is reasonable to suppose the correlation exists in some places.

1. Introduction mixing. In general, different tracers give different tracer ages for the same seawater sample (Doney et al. 1997; In the last two decades there has been an enormous Wunsch 2002; Waugh et al. 2003). Insight on how to expansion in the number of chemical tracer measure- interpret tracer ages has been gained from the recent ments in the ocean. These observations provide valu- development of theories of the distribution of transit able information on ocean ventilation rates, water-mass times (Hall and Plumb 1994; Holzer and Hall 2000; formation processes, and pathways of interior flow (En- Haine and Hall 2002; Deleersnijder et al. 2001). Transit gland and Maier-Reimer 2001 and Schlosser et al. 2001 time distributions (TTDs) capture complete informa- provide useful starting points to the relevant literature). tion about the transport processes in a flow but are not They are often used to calculate so-called tracer ages dependent on any specific tracer source (by transport (e.g., Doney et al. 1997; Robbins and Jenkins 1998), we mean advection and diffusion). Transit time distri- although the interpretation of these ages, as well as of butions have been recently estimated from tracer ob- the tracer field itself, is difficult because of the effects of servations and used to estimate more accurate inven- tories of anthropogenic carbon in the ocean (Hall et al. 2002, 2004; Waugh et al. 2004; Klatt et al. 2002; Stein- Corresponding author address: Dr. Hong Zhang, Department of Earth and Planetary Sciences, The Johns Hopkins University, feldt and Rhein 2004). Insight into the connection be- 301 Olin Hall, 3400 N. Charles St., Baltimore, MD 21218. tween tracers and the ocean circulation has also been E-mail: [email protected] obtained from simulations of idealized and realistic

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JPO2812 NOVEMBER 2005 ZHANG ET AL. 2251 ocean models (e.g., Musgrave 1990; Thiele and derstand the similarities and differences between PV Sarmiento 1990; Khatiwala et al. 2001; Haine and Hall and tracer fields. Potential vorticity is the dynamical 2002; Steinfeldt 2004). master variable governing large-scale low-frequency These studies have improved our understanding of motions in rotating stratified fluids. It may be inverted tracer ages, and thus our ability to interpret tracer data. to yield the velocity and density fields under suitable, Specifically, TTD theory has shown that there are use- common, approximations. Modern theories of the ful relationships between transport time scales and dif- ocean general circulation routinely exploit this prop- ferent tracer ages (Haine and Hall 2002; Hall and Haine erty. Thus they are akin to the theories of passive tracer 2002; Waugh et al. 2003). These relationships are stated fields, although, of course, they are (usually) nonlinear. explicitly in section 3.3 below, but essentially they say For example, theories of the subtropical ocean circula- that for steady flow tracer ages from tracers with simple tion are in this category (Pedlosky 1996). We believe a source functions (exponential or low-order polynomial) synthesis of the dynamical (PV) and kinematic (TTD) can be used to estimate both the mean transit time of theories is possible and would allow a fruitful cross-fertili- the TTD and the characteristic range of transit times in zation of ideas between the two fields. Such a merger is the TTD (its “width”). Characterizing the TTD through beyond our current scope, but by seeking to diagnose, estimating its mean and width is an attractive way to understand, and explain the links between PV and diagnose the integrated effects of advective and diffu- tracer here we are taking a preliminary step toward it. sive transport in the ocean (Waugh et al. 2002, 2004; Our approach is to address these questions using a Klatt et al. 2002; Steinfeldt and Rhein 2004). Of course, barotropic, double-gyre circulation model and simula- we have no way to directly measure the TTD for the tions of several different chemical tracers. The simple real ocean, and even for an ocean circulation model it is system is not intended to represent any specific ocean arduous to calculate. Instead, we have some knowledge basin but it captures several key features of the real of various chemical tracer fields. The relationship midlatitude wind-driven ocean circulation, namely, cy- above is therefore useful because it allows one to di- clonic (subpolar) and anticyclonic (subtropical) basin- rectly infer information about the TTD from the avail- wide asymmetric gyres, which are stronger on the west- able tracer data. An almost universal assumption of ern side of the basin. This model displays a variety of ocean TTD studies to date is that the flow is steady (see solutions, including steady, periodic, and chaotic flows. references cited above). Ocean circulation is unsteady Transitions between these circulations are controlled and chaotic, however, and in fact, as we show here, the by a single parameter related to the Rossby number. relationship above also holds in this case. This result is Several studies investigate the circulation dynamics in important because it is of significant practical value. A this system (Jiang et al. 1995; Meacham 2000; Chang et major goal of this paper is to investigate this relation- al. 2001; Ghil et al. 2002). Models of this type have also ship and its significance. been used to explore idealized tracer dispersion (Mus- Another major goal is to explore the relationship be- grave 1990; Thiele and Sarmiento 1990; Richards et al. tween tracer age and the dynamics of the flow by com- 1995; Jia and Richards 1996; Figueroa 1994; Haine and paring and contrasting tracer age with potential vortic- Gray 2001). For these reasons the barotropic double- ity (PV; see, e.g., Salmon 1998). There are compelling gyre circulation model is well-suited to our study. The reasons to do so. Specifically, PV is advected and dif- focus is on the links between tracer age, PV, and trans- fused in very similar ways to dynamically passive (i.e., port time scales for a chaotic flow with a range of tem- linear) chemical tracer. This useful property is often poral and spatial scales. We seek generic results that are exploited to understand the dynamical evolution of a insensitive to the detailed model configuration and that flow. Potential vorticity and chemical tracers have dif- may be extrapolated to the real ocean. ferent sources and sinks, however, and it is unclear ex- The paper is organized as follows: The circulation actly how their distributions are linked or why. A close model and numerical experiments of tracer transport connection between PV and chemical tracers has been are described in section 2. In section 3 we show results observed in the upper troposphere and stratosphere from these experiments. We focus on the tracer age and (e.g., Danielsen 1968; Leovy et al. 1985; Allaart et al. then examine the connection with PV. A discussion of 1993). Indeed, PV is often used in atmospheric studies these results and their implications is in section 4. as a proxy for chemical tracers (e.g., Schoeberl et al. 1989; Lary et al. 1995; Morgenstern and Marenco 2000). 2. Model and numerical experiments Yet the extent of these connections in the ocean is poorly known. We solve the equations for a barotropic double-gyre There is another, more fundamental, reason to un- circulation on a midlatitude beta-plane forced by a

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steady zonal wind stress. Here, we describe the dynami- TABLE 1. Model parameters. cal model (section 2a) and then the tracer experiments Parameter Value performed in this circulation (section 2b). ϭ ϩ ␤ ϭ Ϫ4 Ϫ1 Coriolis parameter f(y) f0 yf0 10 s ␤ ϭ Ϫ11 Ϫ1 Ϫ1 a. Dynamical model 10 s m Water depth H ϭ 2500 m ␯ ϭ 2 Ϫ1 The governing equations for the circulation model Viscosity coefficient h 100 m s Water density ␳ ϭ 997 kg mϪ3 are 0 Gravitational acceleration g ϭ 9.8msϪ2 ϭ Ϫ2 Du Ѩ␩ X Wind stress X0 0.1Nm Ϫ ␷ ϩ Ϫ ␯ ٌ2 ϭ ͑ ͒ ϭ f g Ѩ h hu ␳ , 1 Domain extent L 1200 km Dt x 0H ␦ ϭ ϭ Inertial boundary layer thickness IL 0.027L 32 km ␦ ϭ ϭ D␷ Ѩ␩ Munk boundary layer thickness ML 0.018L 22 km ϩ ϩ Ϫ ␯ ٌ2␷ ϭ ͑ ͒ Rossby deformation radius L ϭ ͌gH/f ϭ 1600 km fu g Ѩ h h 0, and 2 R 0 Dt y Grid resolution ⌬x ϭ⌬y ϭ 20 km ⌬ ϭ D␩ Ѩu Ѩ␷ Time step t 20 min ϩ Hͩ ϩ ͪ ϭ 0, ͑3͒ Dt Ѩx Ѩy where (u, ␷) are the (x, y) (east, north) components of Hopf bifurcation then leads to periodic asymmetric the fluid velocity, ␩ is the surface displacement, f(y) ϭ flow, before chaos finally sets in through a period- ϩ ␤ f0 y is the Coriolis parameter, H is the uniform doubling cascade (Jiang et al. 1995; Meacham 2000; ␯ ␳ water depth, h is the (eddy) viscosity, 0 is the water Chang et al. 2001; Ghil et al. 2002). The focus here is on density, g is the gravitational acceleration, and X is the asymmetric, chaotic solutions, and experiments in other zonal wind stress. The Lagrangian time derivative is regimes are reported elsewhere. D/Dt. The wind stress is assumed to have a symmetric, We characterize the spatial and temporal variability ϭ ␲ constant, form: X(y) X0 sin( y/L), where L is the of the flow by examining ␩ [sea surface height (SSH)] width of the square domain. The boundary conditions and PV fields. Potential vorticity is (␨ ϩ f )/(H ϩ ␩), are no-slip at the solid sidewalls; (u, ␷) ϭ 0atx ϭ (0, L) where ␨ is the relative vorticity and f is the planetary and y ϭ (0, L); and free-slip at the seafloor. Table 1 vorticity. Here, PV is given accurately by q ϭ (␨ ϩ f )/H

shows the values of the model parameters used in the because the Rossby deformation radius, LR, exceeds simulations below. the domain size (Table 1). The dynamics of this wind-driven model is controlled The governing equation for PV is by two nondimensional parameters (Pedlosky 1996): Dq ͒ ͑ the nondimensional inertial and frictional Munk layer Ϫ ␯ ٌ2 ϭ h hq Sq, 5 thicknesses, Dt Ϫ Ϫ ϭϪ ␤ 1 ␲ ץ ץր ր ϭϪ␳ 2 1 1 2 ␯ 1 3 where Sq ( 0H ) X/ y U H cos( y/L)is U h ␦ ϭ ͩ ͪ and ␦ ϭ ͩ ͪ , ͑4͒ the PV source due to the wind stress. In the absence of I ␤L2 M ␤L3 forcing and dissipation, PV is conserved along stream- where U is the characteristic (gyre interior) flow speed. lines. The no-slip sidewalls provide a flux of relative These two parameters measure the relative importance vorticity (and hence q) given by k ϫ F, where k is the of nonlinearity (by the ratio of relative vorticity to the unit vertical vector and F is the frictional acceleration at contrast in planetary vorticity) and lateral dissipation the wall (Marshall et al. 2001). In the present case, the (by the ratio of viscous spindown rate to the contrast in western boundary provides cyclonic (anticyclonic) vor-

planetary vorticity). They are related to the Rossby ticity to offset Sq in (5) in the subtropical (subpolar) number (Ro ϭ U/␤L2) and the horizontal Ekman num- gyre. No exact slow dynamical balance exists in (1)–(3) ϭ ␯ ␤ 3 ␦ ϭ 1/2 ␦ ϭ 1/3 ber (Eh h/ L ) through I Ro and M Eh . because the system supports relatively fast inertia- The velocity scale U and the characteristic wind stress gravity waves. Nevertheless, PV is the central dynami-

X0 are linked by the Sverdrup balance. For a square cal variable because solutions typically evolve in geo- basin with a double-gyre wind stress the scaling leads to strophic balance without exciting these modes. We also ϭ ␲ ␳ ␤ U X0/( 0 LH). characterize the flow with two scalar diagnostics: The This system displays several types of solutions. They basin-averaged kinetic energy (KE) and the gyre trans- range from steady flow with symmetric gyres to chaotic port difference (GD) between the two gyres, measured ␦ ϭ |␩ | Ϫ |␩ | ␩ ␩ flow. As the velocity increases (rising I through in- using SSH: GD tr po , where tr and po are the

creasing X0 or decreasing H), the symmetric, steady maximum and minimum SSH within the subtropical |␩ | ϭ ␩ |␩ | ϭ | ␩ | solution loses symmetry at a pitchfork bifurcation. A and subpolar gyres [ tr max( ), po min( ) ;

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ϭ i Chang et al. (2001) used a similar diagnostic]. The GD TABLE 2. Tracer experiments. The total tracer source is Sc Sc ϩ b Ϫ Ն is a measure of a solution’s symmetry with respect to [Sc(t) C]/T, where the second term only applies for y 1180 ϭ the midbasin axis (GD ϭ 0 corresponds to symmetric km. Time scale T 1 day controls the rate of relaxation in the northern boundary region. gyres). i b ␶ The system (1)–(3) is integrated numerically using a Expt Sc Sc(t) Tracer age, Parameter simple configuration of the Massachusetts Institute of Ideal age 1 0 C Technology general circulation model (MITgcm; Mar- Radioactive Ϫ␭C 1 ␭Ϫ1 ln CϪ1 ␭ ϭ 0.028 yrϪ1 shall et al. 1997). We use a spatial grid step of 20 km tracer Ϫ ϭ Ϫ1 and a time step of 20 min. The focus is on stationary Linear transient 0 a1ttC/a1 a1 1yr Exponential 0 exp␭ttϪ (lnC)/␭␭ϭ 0.028 yrϪ1 solutions to these equations so we discard the dynami- transient cal transients as the flow accelerates from rest. Typi- 2 Ϫ ͌ ϭ Ϫ2 Quadratic 0 a2t t C/a2 a2 1yr cally we integrate the model for 100 yr once this ad- transient justment is complete. where ␭ is the radioactive decay constant of the tracer. b. Tracer model This simple form is convenient for the present pur- We simulate five passive tracers in the circulation poses, but a slightly different expression involving the model described above. In each case, the tracer con- ratio of parent to daughter concentrations is also com- centration C satisfies mon. Last, for conserved tracers with monotonically increasing boundary concentration, tracer age is de- DC fined as the elapsed time since the boundary concen- ͒ ͑ Ϫ ␯ ٌ2 ϭ c hC Sc , 6 tration was equal to the interior concentration. That is, Dt ␶ Ϫ ␶ ␭ Ϫ ␶ Ϫ is such that C equals a1(t ), exp (t ), and a2(t ␶)2 for the linear, exponential, and quadratic transient where S is a source/sink term and ␯ is the diffusion c c tracers, respectively. Again, more complex formulas coefficient. We pick ␯ ϭ ␯ , that is a Schmidt (or h c are possible, such as a tracer age depending on the Prandtl) number, ␯ /␯ , of 1. This choice is reasonable h c concentration ratio of two transient tracers, but they because the viscous and diffusive processes represented are not needed here. The tracer experiments are per- by ␯ and ␯ are unresolved turbulent motions. The h c formed by integrating (6) in the MITgcm using a cen- molecular Schmidt number for real dissolved chemical tered second-order advection scheme. tracers is O(10–100), however, which will likely have a substantial impact on the correlation between PV and 3. Numerical results tracer ages near the molecular dissipation scale. The tracer experiments are set up as follows (see also In this section we describe results from numerical Table 2): To crudely represent the strong ventilation of experiments using the model explained in section 2. the convective subpolar gyre of the North Atlantic First, we present dynamical fields to characterize the ocean, we specify a tracer source or sink near the north- circulation (section 3a). Then, we discuss the corre- ern boundary (for y Ն 1180 km; Table 2). This source sponding ideal age results (section 3b). In section 3c the ensures rapid relaxation to a specified, time-dependent relationship between the various tracer ages is ex- concentration and has been used in several previous plored. Section 3d shows how the potential vorticity studies (e.g., Musgrave 1990; Thiele and Sarmiento and ideal age fields are related. 1990; Doney et al. 1997; Robbins and Jenkins 1998). a. Dynamics No-flux tracer boundary conditions apply. We simulate the “ideal age” tracer (Thiele and Sarmiento 1990; En- A series of 100-yr simulations has been performed gland 1995), which has a unit source at all places and using the model parameters listed in Table 1 with fluid times. We also simulate a radioactive decaying tracer depth H varying from 6000 to 2500 m. This corresponds ␦ ϭ ␦ with a constant northern source concentration. Last, we to fixing M 0.018 and varying I between 0.017 and simulate three conserved tracers that have transient his- 0.027. Chang et al. (2001) performed a very similar se- tories in the northern source region: a linear, exponen- ries of simulations, although they used free-slip side- ␦ ϭ tial, and quadratic increasing tracer. walls and fixed M 0.0246. The flow regimes and From the tracer concentrations in these experiments transitions in our simulations are qualitatively similar to we derive tracer ages, ␶ (see also Table 2). The ideal age their results although the bifurcations occur at slightly ␦ is given by the concentration of the ideal age tracer different I. Here we focus on the chaotic flow with ␭Ϫ1 Ϫ1 ϭ ␦ ϭ itself. For the radioactive tracer, age is lnC , H 2500 m ( I 0.027). We also integrate the tracer

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Ϫ FIG. 1. Time series of (a) average kinetic energy per unit mass in each model grid cell (KE; m2 s 2), and (b) gyre transport difference (GD; m: see section 2). The stars in (a) indicate the times of the snapshots shown in Fig. 2. equation using the time-mean circulation from this fluctuations (years 85–90; Fig. 2). Between years 83 and case. By comparing steady and unsteady flow we quan- 85, KE increases as the flow becomes stronger. The tify the effect of unsteadiness on tracer age fields and extrema of SSH for the subtropical and subpolar gyre the underlying transport. also increase [SSH is a good surrogate for streamfunc- In the chaotic circulation, KE and GD fluctuate ir- tion because the Rossby number is small, O(10Ϫ3), and regularly (Fig. 1 shows their temporal evolution). The hence the flow is very nearly geostrophic]. For example, principal KE oscillations have a period of 1–2 yr. The the flow at 83.8 yr has high KE (8.3 ϫ 10Ϫ3 m2 sϪ2) and flow is asymmetric and characterized by strong mean- extrema of SSH at Ϯ0.45 m. The two gyres are almost dering of the intergyre jet with several (small) vortices symmetric. From this time to the peak KE at 85 yr, the present. There are also periods of larger and longer KE inertial recirculation gyres expand and the circulation oscillations, however. Then, the flow is characterized by accelerates. Peak SSH in each gyre increases to 0.57 strong, near-symmetric basin-scale dipoles, a long zonal and Ϫ0.51 m by year 85.4, shortly after the peak in KE jet, and weak meandering. Variations in GD are at (Fig. 2a). Around this time, the flow is dominated by higher frequency than those in KE: typically their pe- nearly symmetric cyclonic and anticyclonic gyres. Fig- riod is around 50 days. The spells of large, slow KE ure 2d shows the PV at this time. There are two regions oscillations, and their breakdown, are associated with of uniform PV that fill the gyre centers; one with rela- relatively large amplitude GD variability. These results tively low PV (4 ϫ 10Ϫ8 mϪ1 sϪ1) in the south and one are generally consistent with previous work by Mea- with high PV (4.4 ϫ 10Ϫ8 mϪ1 sϪ1) in the north. Like cham (2000) and Chang et al. (2001). SSH, the PV field is nearly symmetric. The jet flows We now examine the flow as it approaches the peak eastward, parallel to the SSH and PV fronts between of a large KE excursion then returns to smaller, faster the two gyres.

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Ϫ Ϫ FIG. 2. Model fields of (a)–(c) SSH, ␩ (0.1 m), light shades are ␩ ϽϪ0.25 m, dark shades are ␩ Ͼ 0.25 m; (d)–(f) PV, q (10 8 m 1 sϪ1), light shades are q Ͻ 4.15 ϫ 10Ϫ8 mϪ1 sϪ1, dark shades are q Ͼ 4.35 ϫ 10Ϫ8 mϪ1 sϪ1; and (g)–(i) ideal age, ␶ (yr), light shades are ␶ Ͻ 15 yr, dark shades are ␶ Ͼ 17 yr, at model times 85.4, 86.1, and 89.3 yr, respectively. Figure 1a shows the times of these snapshots. They are at the peak, decay, and equilibration of the large excursion in kinetic energy around 85 yr.

By 86.1 yr the flow has changed significantly, al- uniform PV are being broken apart as the gyres col- though the KE remains relatively high (Fig. 2b). The lapse. The central jet now meanders substantially and is southern, anticyclonic gyre is intruding into the north- weaker although it still penetrates into the eastern half ern half of the domain at the expense of the cyclonic of the domain. gyre there. A weaker anticyclonic vortex remains in the The circulation and the KE continue to diminish until southwest part of the domain and GD oscillates at large around year 90. Figure 2c shows the flow pattern with a amplitude. The cyclonic subpolar gyre has diminished relatively low KE (1.7 ϫ 10Ϫ3 m2 sϪ2) at 89.1 yr. The and moved slightly to the southwest. The PV field re- gyres are now much weaker (peak SSH is Ϯ0.15 m), flects the distortion by the flow (Fig. 2e). The pools of restricted to the western third of the domain, and are

Unauthenticated | Downloaded 09/25/21 03:30 PM UTC 2256 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 35 asymmetric. The fluctuations in GD are smaller than nificantly (Fig. 2h). The weakening cyclonic gyre has before. The PV field shows several isolated extrema given way to northward intrusions of high ideal age associated with the circulation evident from the SSH (high SSH and low PV). The intergyre front has moved field (Fig. 2f). In the eastern half of the domain the PV south in the western half of the domain carrying contours are oriented zonally because of the slow cir- younger fluid. The peak ideal age remains around 20 yr, culation there and the underlying meridional gradient however. At year 89.3 the area of strong ideal age gra- in planetary vorticity. The region of strong relative vor- dients has receded to the west (Fig. 2i). Meridional ticity (and hence PV distortion from east–west con- ideal age contours now fill most of the eastern half of tours) has receded to the west as the circulation slack- the basin. To the west, small pools of uniform ideal age ens. The jet is now too weak to penetrate far into the are associated with extrema in SSH, but they are domain. smaller, and 1–2 years younger, than at the time of most Although details differ, this genesis and decay of energetic flow. strong symmetric gyres is characteristic of all the high Clearly there is rich time variability in this circula- KE episodes in Fig. 1. In some cases, breakdown in- tion. To emphasize this point, we show time-averaged volves a weakened subtropical gyre, rather than a fields in Fig. 3. The mean circulation is very nearly weakened subpolar gyre and, among the few examples symmetric as revealed by the SSH and PV fields (Figs. in Fig. 1, there is no evidence of a preferred sequence. 3a,b; GD ϭϪ0.02 m). On average the flow is organized In any event, the western half of the domain experi- into two western-intensified gyres of equal amplitude. ences intermittent, energetic flow that involves large The mean circulation strength is typical of the circula- departures from a simple symmetric double-gyre pat- tion during the normal low-energy regime shown in Fig. Ϫ Ϫ tern. In contrast, the eastern half of the domain is much 2c (KE ϭ 3.3 ϫ 10 3 m2 s 2). The pattern of circulation less active. It can be significantly perturbed by the flow resembles the high-energy phase of Fig. 2a, however. from the west, but is typically quiescent. Small pools of uniform PV are present in the gyre centers, but the flow is too weak to distort the PV field far from the meridional trend in planetary vorticity. The b. Ideal age time-averaged ideal age field (Fig. 3c) does not look We now examine the ideal age field for the sequence like any of the snapshots in Figs. 2g–i (the same is true of flows described above. Ideal age is a robust estimate for PV). There are large areas of uniform ideal age in of the characteristic transport time scale. It is equal to the western half of the domain, but, in the eastern half, the mean transit time in both steady and unsteady flow the ideal age contours turn north as the field ramps up (Haine and Hall 2002; see also section 3c). The steady to the eastern boundary. There is a boundary layer in case has received much attention (see references in sec- ideal age along the northern wall and northern half of tion 1) but the unsteady case has not been explored the western wall. before. The impact of the time variability on ideal age is seen Figures 2g–i show three snapshots of the ideal age in Fig. 3d. This diagram shows the ideal age field in the field, corresponding to the peak, decay, and equilibra- time-averaged circulation itself. That is, we solve (6) for tion of the high-energy event at year 85 (Fig. 1). At time ideal age using the mean flow shown in Fig. 3a. The 85.4 yr, the ideal age in the western half of the domain unsteadiness has two principal effects (comparing Figs. consists of two pools of uniform values (Fig. 2g). The 3c and 3d). First, it keeps the ideal age 4–12 years subpolar pool has ideal age of 15 yr and the subtropical younger than it would otherwise be. Second, it prevents pool has ideal age of 19 yr. There is a steep drop to isolated ideal age extrema from forming in the gyre vanishing ideal age at the northern boundary, consis- centers. For example, the oldest values in the mean tent with the strong sink there (section 2b). The low ideal age field are near 19 yr at the southern boundary. ideal age is carried south along the western boundary The greatest ideal age values from the mean circulation following the geostrophic flow (Fig. 2a). To the south, are near 30 yr, however, in the middle of the cyclonic there is a local minimum in ideal age, then a weak gyre (Fig. 3d). Cox (1985), for example, reports similar increase to the boundary. The ideal age is being results in a simulation of midlatitude thermocline ven- dragged by the anticyclonic circulation that peaks in tilation. strength at this time. In the eastern third of the domain Circulation unsteadiness changes the character of the the ideal age contours run north–south, increasing to tracer field fundamentally. Perhaps most significant is the wall. A strong gradient in ideal age exists at the the following fact: The flow variability causes the align- location of the eastward intergyre jet. ment of the ideal age contours to twist by Ϯ90°, so that By time 86.1 yr the ideal age field has evolved sig- the mean streamlines (Fig. 3a) are perpendicular to the

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Ϫ Ϫ Ϫ FIG. 3. Time-averaged model results. (a) Mean SSH (0.1 m), (b) mean PV (10 8 m 1s 1), (c) mean ideal age (yr), and (d) ideal age (yr) in the time-averaged circulation [i.e., (6) is solved with the ideal age source (Table 2) in the circulation shown in (a)]. The period of time averaging is from year 50 to year 100. The location of the time series of tracer age shown in Fig. 4 is marked with the southern star on (c) and (d). Fig. 7 shows time series of ideal age and PV at the stars. mean ideal age contours in the western subtropics (Fig. ideal age field differs qualitatively from all the other 3c). The ideal age contours from the time-averaged ideal age maps where crossing of the streamlines by age flow are parallel with the streamlines, however (Fig. contours is almost nowhere seen. 3d). This alignment is also true in the eastern half of the basin, where the flow variability is weak. Interestingly, c. Relationship between tracer ages the instantaneous ideal age and SSH fields in Fig. 2 also We now examine the relationships between various show alignment, even though they are crossed in the tracer ages. Whenever mixing occurs tracer ages from time average. Hence, there is no connection between different tracers diverge. The reason is there is a range alignment in the instantaneous field and alignment in of transit times in the TTD in the presence of mixing. the time-average field. In this respect, the time-average Any single time scale cannot capture this range of tran-

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sit times. Different tracer ages therefore do not coin- width ⌬ that is 14.6 Ϯ 2.0 yr. We may check this value cide as they depend on the TTD in different ways. in two ways: First, the TTD itself has been computed Nevertheless, some simple relationships exist be- for the time-averaged circulation and gives (⌫, ⌬) ϭ tween the tracer ages and the TTD. Specifically, see the (29.6, 16.4) yr for the location in Fig. 4, in good agree- following. ment (the TTD will be shown in detail elsewhere). Sec- ⌬ 1) The ideal age equals the mean transit time ⌫. [The ond, property 4 can be used to find by considering a mean transit time equals the first moment of the tracer with a quadratic source function. The concentra- TTD and is the tracer age derived from the concen- tion (or the tracer age) then depends only on the first tration of a conserved transient tracer with a linearly two moments of the TTD (see the appendix for details). ⌬ϭ increasing source; Haine and Hall 2002, their (17).] Simulation of a quadratic tracer gives 16.4 yr, again 2) A conserved tracer with an exponentially growing in good agreement. This verifies property 4, at least for ϭ ϭ source (at rate ␭) yields a tracer age that equals the the case N 2 (note that the case N 1 simply tracer age from a tracer with a constant source, amounts to the familiar statement that linear tracer age which decays exponentially with rate ␭. equals ⌫; Haine and Hall 2002). Results from other 3) The tracer age of an exponentially growing tracer, locations yield similar results to those shown in Fig. 4, ␶␭, is approximately although they typically converge faster because the mean age at the site shown is near the maximum any- ␶ Ӎ ⌫ Ϫ ⌬2␭ ͑ ͒ ␭ 7 where in the domain (Fig. 3d). if ⌬␭ Ӷ 1, where ⌬ is the width of the TTD (i.e., We now consider these properties for the unsteady ⌬2 ϭ ␮ ␮ 2 2, where 2 is the second centered moment circulation described above. Figure 4b shows the time of the TTD) and successive higher moments of the series of tracer ages for the unsteady flow. The most TTD do not diverge too strongly. obvious difference between these results and the steady 4) The tracer age of a conserved tracer with a polyno- case is the secular variability of the tracer ages in un- mial source function of degree N depends on the steady flow. Perturbations in different tracer ages are in first N moments of the TTD only. phase and of very similar magnitude. Clearly, they are caused by the evolving tracer fields as we see for ideal Relationship 1 is well-known for steady flow (Khati- age in Fig. 2. The typical tracer ages are also signifi- wala et al. 2001; Steinfeldt 2004; Hall and Plumb 1994). cantly lower as discussed in section 3b above. Other- It also applies for unsteady flow [Haine and Hall 2002, wise, the two diagrams are remarkably similar. Ideal their (17)], although we are unaware of any studies that exploit this fact. Relationships 2–4 are demonstrated age overlies tracer age from the linear tracer, verifying for steady flow by Waugh et al. [2003; see also Delhez property 1 applies to unsteady flow. Exponential and et al. (2003) for relationship 3]. In fact they hold for radioactive tracer ages also coincide, verifying property unsteady flow too and the proof for each case is in the 2 applies to unsteady flow. Property 3 is tested in two ␶ appendix. We now confirm and illustrate these relation- ways: First, we show on Fig. 5 the average values of ␭ ships in our unsteady flow. once the ages reach a statistical steady state [the calcu- ␭Ϫ1 First we confirm that the theory for our tracer simulations last at least O( ) years to become stationary]. lations holds in steady circulation. We use the time- Again, there is a linear relationship between average averaged flow shown in Figs. 3a and 3b and simulate the exponential tracer age and decay rate. The intercept at suite of tracers listed in Table 2. Figure 4a shows the vanishing ␭ is 17.2 yr, which agrees very well with the time series of different tracer ages at a location in the time-average ⌫ (i.e., the first moment of the time- southern anticyclonic gyre (see Fig. 3d). Property 1 is averaged TTD) of 17.4 yr, derived by averaging the demonstrated by the fact that the linear tracer age linear tracer age over years 50–100 (Fig. 4b). The time- (equal to ⌫) overlies the ideal age on Fig. 4a. Property average TTD width is 10.7 Ϯ 1.1 yr. Property 3 can also 2 is demonstrated by the fact that the exponential and be tested in a stronger form at every individual time, radioactive tracer ages overlie each other. Property 3 is rather than averaged over time. To do so, we fit ␶␭ explored in Fig. 5. Here, we plot tracer age, ␶␭, from the against ␭ for each individual time, as in Fig. 4. Again, radioactive tracer against decay rate for five different linear fits are accurate. The intercepts give estimate for ␭ ⌫ ⌫ values of . The data lie close to a straight line that (denoted est), which compare well with the values intercepts ␭ ϭ 0at␶␭ Ӎ 29.3 yr. The value of ⌫ from Fig. from the linear tracer age. Figure 6 shows these two 4a is 29.5 yr, in excellent agreement. This concordance, time series and their excellent agreement confirms and the linear relationship, verifies property 3 [(7)]. property 3. Last, property 4 can be tested for the case N The slope of the line yields an estimate of the TTD ϭ 2 by simulating a quadratic tracer to infer ⌬. This

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FIG. 4. Time series of tracer ages ␶ at the location indicated by the southern star in Fig. 3 using (a) time-averaged flow and (b) chaotic flow.

⌬ value is compared with the estimate (denoted est) tions for tracer and PV are so similar (section 2). It is based on the slopes of the linear fits of ␶␭ to ␭ explained unclear exactly how they are linked, however, because above. Figure 6 shows good agreement again (at least the sources and sinks are different. Here, we quantify ⌬ once est has stabilized at around 150 yr), consistent this connection by analyzing the linear correlation be- with both properties 3 and 4. The average value of ⌬ is tween the PV and ideal age fields. 11.6 Ϯ 0.5 yr, consistent with the estimate based on Fig. In Fig. 7 we show the ideal tracer age and PV time 5 above. Interestingly, variability in ⌫ is much stronger series from a site in the subtropics and in the subpolar than variability in ⌬ for this circulation (it is also cor- gyre (marked on Fig. 3). In the subtropics there is a related with coefficient 0.80 for years 150–200). This is strong link between the variability in ideal age and PV. the reason that the linear tracer age and the exponen- High PV anomalies almost always coexist with low tial ages in Fig. 4b are so closely related. The offset in ideal age anomalies (including the high KE episode dis- the time series is given, via property 3, to be approxi- cussed in section 3a). This relation may also be seen in mately ⌬2␭. In this case, the product is 2.9 Ϯ 0.6 yr Figs. 2e, 2h, and Fig. 3, for example. In the subpolar (using the time-average value of ⌬), which agrees very gyre, there is no clear relationship, however. The only well with the offset of 2.9 yr in Fig. 4b. obvious difference between the series at the two locations is that the amplitude of ideal age and PV variabil- d. Relationship between potential vorticity and ity is about one-half as strong in the subpolar gyre as it ideal age is in the subtropics. The PV and ideal age fields are closely related in Fig. In Fig. 8 we investigate the ideal age and PV relation 2. Close links are expected because the governing equa- in more detail. It shows the temporal correlation coef-

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subpolar gyre and eastern half of the domain, the correlation is weaker and of either sign. Although it does not exceed the threshold for statistical significance, correlation in the latter region possesses some spatial structure. Namely, the correlation contours bend round the central western region of strong flow variability in a coherent, organized way. We show in Fig. 9 ideal age/PV correlations from three other experiments with different configurations. In Fig. 9a the flow is identical to the standard case (Fig. 8), but the tracer source region is extended south to cover the northern quarter of the domain (i.e., for y Ͼ 900 km). This difference causes the transport time scales to decrease because of closer proximity to the source region. The region of highly significant ideal

FIG. 5. Tracer age, ␶␭, of exponential tracer with a source that age/PV correlation now covers most of the domain and grows at rate ␭ vs ␭. Results from both steady (stars) and chaotic the pattern of correlation contours closely resembles flow (triangles) are shown. For the chaotic flow, average values of those in Fig. 8. Figure 9b shows an experiment identical ␶ ␭ are shown once the age has reached a statistical steady state. to the standard case, but with a depth H of 3500 m, not Straight lines indicate linear best fits to the model data in each case. The intercept estimates ⌫ and the slope estimates Ϫ⌬2 from 2500 m. The circulation in this case is still chaotic, but property 3 (see section 3c). These results are from the location with much reduced variability (the velocity scale U is ␦ shown in Fig. 4. decreased by 40% and I is 18% lower). This solution makes aperiodic transitions between relatively long- lived states and the regions of well-mixed PV are ficient of the ideal age and PV over years 50–100. Cor- greatly cut back. Now the area of highly significant correlations with absolute value greater than 0.5 (signifi- relation shrinks somewhat, but the organized curving cant at the 99.9% confidence level) are shaded. There is bands of correlation in the east are more pronounced significant anticorrelation in the western subtropical and significant than in Fig. 8. Figure 9c is for a domain gyre and midbasin region, as seen in Fig. 7. These areas that is twice as wide as the standard experiment, but do not obviously coincide with features in the time- with no wind stress applied to the middle half of the averaged ideal age or PV fields (Fig. 3). In the central basin (i.e., for 600 km Ͻ x Ͻ 1800 km). This experiment changes the PV dynamics in a fundamental way because the unforced fluid materially conserves PV, exactly like inert tracer. Significant correlation exists in a smaller fraction of the (bigger) domain in this case. The correlation is still mainly found in the western part of the subtropical gyre but significant correlation extends into the central unforced half of the basin. These experiments confirm the findings of the standard experiment: ideal age and PV are highly correlated in some, but not all, places. The correlation structure depends somewhat on the details of the tracer source (Fig. 9a), the character of the flow (Fig. 9b), and the PV dynamics (Fig. 9c). The presence of the highly significant ideal age/PV correlation is insensitive to these factors, however—the presence of the correlation itself is a generic result in these experiments. Results from other experi- ⌫ ⌫ ␦ FIG. 6. Time series of from linear tracer age, est from inter- ments with greater H (smaller I), or a different tracer cepts of linear fits to ages from exponential tracers with various source, show similar results. rates, ⌬ from a quadratic transient tracer and ⌬ from slopes of est What is the explanation of this subtle correlation linear fits to ages from exponential tracers with various rates. ⌫ ⌫ structure? To address this question we consider the Coincidence of with est is evidence to support property 3 while ⌬ ⌬ sources and sinks of ideal age tracer and PV in the coincidence of with est supports properties 3 and 4 (see section 3c). These results are from the location shown in Fig. 4. standard experiment (see section 2). Recall that PV has

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FIG. 7. Typical time series of (a) PV in the subtropics, (b) PV in the subpolar gyre, (c) ideal age in the subtropics, and (d) ideal age in the subpolar gyre. The sites of these time series are shown by stars on Fig.3.

a latitude-dependent source, Sq, that is negative in the tortions to the mean circulation due to unsteadiness southern half and positive in the northern half of the thus cause anticorrelated PV and ideal age variations domain. Potential vorticity is reset by friction at the because both tracers are materially conserved (and dif- western wall. There is, therefore, no meridional asym- fusive fluxes are relatively weak). This anticorrelation metry in the PV (or SSH) statistics, as evident in Figs. is seen in Figs. 7 and 8. 3a and 3b (␩/H Ϸ 10Ϫ4 and is too weak to cause no- In the subpolar gyre, however, the ideal age bound- ticeable symmetry breaking). The ideal age tracer has ary layer at the northern and western walls is much substantial meridional structure, however. The ideal more important. Although SC tends to increase ideal age source, SC, is unity everywhere outside a narrow age along trajectories, diffusive fluxes drain the subpo- strip at the northern wall where the ideal age is reset to lar gyre of ideal age tracer. Stirring by the variable zero (Table 2). In consequence, the time-averaged ideal circulation mediates this diffusive exchange as can be age field is qualitatively different in the centers of the seen in the detached boundary layers in Figs. 2g–i. In subtropical and subpolar gyres (Fig. 3c). the gyre center, where flow variability is largest, the In the subtropical gyre fluid following mean stream- time-averaged ideal age field is therefore nearly flat. lines loses PV along its trajectory because of Sq (trajec- Fluid following mean streamlines in this area tends to tories cross mean PV contours; Figs. 3a,b). Along these maintain constant ideal age on average. Mean PV in- paths, fluid also gains ideal age (because of SC: trajec- creases, however (Sq is positive; Fig. 3b), and the linear tories cross mean ideal age contours; Figs. 3a,c). Dis- correlation between PV and ideal age is thus broken.

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ably, the alternating bands of positive and negative correlation are clearest in Fig. 9a because the basin modes capture the greatest share of the variability in that case. Last, we consider briefly the spatial correlations between PV and ideal age. Examining transects of ideal age and PV from instantaneous fields (e.g., Fig. 2i) re- veals various (spatial) correlation coefficients. In general, zonal transects in the subtropical gyre show significant anticorrelation, but in the subpolar gyre there is insignificant correlation. For instance, a zonal section at y ϭ 400 km gives a highly significant correlation of Ϫ0.86, but a zonal section at y ϭ 800 km gives an insignificant correlation of Ϫ0.1. In both cases, the transects pass through regions of highly significant and insignificant (temporal) correlation (Fig. 8). Discover- ing the boundaries of these areas using measurements that are dense in space but sparse in time—for example, using infrequent ship-based hydrographic sections—is thus difficult. The essential problem lies in the fact that FIG. 8. Pointwise linear correlation coefficient between ideal the temporal correlation is not uniform. age and PV for years 50–100 calculated from monthly instantaneous fields. The contour interval is 0.1 with negative (positive) correlation shown by dashed (solid) contours. Dotted lines indi- 4. Discussion cate zero correlation, and statistically significant correlation is shaded. We have examined the transport in simple barotropic double gyre circulations using several tracers, including In the eastern half of the domain correlation is low potential vorticity. Here we summarize our main find- for a different reason. There, the mean flow is very ings, identify their limits, and discuss the implications weak and ideal age increases to the eastern wall. Flow for the real ocean. variability is also very weak, so the tracer boundary Results show that for steady circulation, tracer ages layer at the northern wall is restricted to a narrow, are consistent with tracer transport theory, as expected. sharp diffusive front. This layer does not detach and stir More important, tracer age relationships that are well into the interior (Figs. 2g–i). For most of the eastern understood in steady circulation also hold in unsteady half of the domain, PV and ideal age contours are flow (properties 1–4 in section 3c; the theoretical ex- therefore near perpendicular. Random distortions of planation is in the appendix). Accurate estimates of the these fields do not result in any systematic relationship mean transit time ⌫ and the width ⌬ of the TTD can between PV and ideal age anomalies. therefore be found from tracers with simple source The origin of the coherent, but weak, correlation in functions (exponential or low order polynomial). Ex- the eastern part of the domain (e.g., in Fig. 9a) is not plicit simulation of the TTD itself is not required. These completely clear. Potential vorticity animations show facts are immediately useful to researchers who study this region contains westward-propagating planetary transport in ocean circulation models using passive waves (also evident in Figs. 2d–f). Periodic oscillations tracers. in the same direction can cause correlated variability in These results are also encouraging for the interpre- PV and ideal age tracer even if the fields are crossed as tation of chemical tracer fields in the real ocean and they are in the east. This effect likely contributes to the illuminate that important practical task. To the extent curved correlation contours seen in Fig. 8. Indeed, spec- that available oceanographic tracers can be described tral analysis of the SSH variations in the eastern part of by simple source functions (exponential or low-order the standard experiment shows power at periods of 55– polynomial), properties 1–4 allow us to estimate both ⌫ 65 and 80–85 days. Linear theory predicts the periods of and ⌬ from the natural oceanic TTD. Radioactive trac- the gravest two standing planetary waves to be at 55 ers, such as carbon-14, argon-39, tritium, and helium-3, and 86 days (Willebrand et al. 1980). From this good have highly reliable and well-known exponential agreement we conclude that the coherent eastern cor- sources (or sinks) away from the ocean surface. Apply- relation is due to these so-called basin modes. Presum- ing the properties discussed here to these tracers is

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FIG. 9. As for Fig. 8 but (a) for a tracer source region that occupies the northern quarter of the domain (i.e., for y Ͼ 900 km); (b) for a depth of H ϭ 3500 m, not 2500 m; (c) for a domain 2 times as wide with no wind forcing in the middle half of the domain (i.e., no forcing for 600 km Ͻ x Ͻ 1800 km); and the contour interval is 0.1 with negative (positive) correlation shown by dashed (solid) contours. Dotted lines indicate zero correlation and statistically significant correlation is shaded. The tracer source region is shown with hatch marks. made harder by the complicated surface sources (e.g., sources are well fit by low-order polynomial or expo- from nuclear weapons tests) or by sparse data, how- nential functions but prior to this phase they are not. ever. Biologically active chemical tracers, such as dis- Exactly how these factors will confound the determina- solved oxygen, have uncertain, variable, interior tion of ⌫ and ⌬ is not clear and deserves further study. sources or sinks but they do have relatively simple sur- Note, finally, that in estimating ⌫ and ⌬ for transport in face sources. Anthropogenic transient tracers, such as lakes and the mode waters of the North Atlantic chlorofluorocarbons, possess relatively smooth (in (Waugh et al. 2002, 2004), we explicitly assumed steady space and time), well-known, surface sources and are transport and a specific functional form for the TTD. inert (or weakly destroyed) in the ocean interior. Dur- The present results suggest that neither of these unre- ing the period of their transient growth the tracer alistic assumptions is actually restrictive.

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The almost identical fluctuations in tracer ages seen travel through the ocean, this would be an important in Fig. 4 may also be of practical importance. They prospect. suggest that time variations in transient tracer ages in The circulation we use is simple, convenient, and in- the real ocean are very closely related in amplitude and expensive but it is qualitatively realistic in important phase (assuming, reasonably, that the tracers have the ways. In particular, the circulation variability is inho- same source regions). Variations in one tracer age mogeneous and anisotropic, just as in the real ocean. likely provide accurate variations in other tracers. Per- Although the statistics are stationary, the variability haps it is possible for variations in dissolved oxygen spans a wide range in time, from O(10) days to O(10) concentration to stand for variations in chlorofluoro- years. Rare, intermittent events play an important role carbon concentration, for example, which is much and the rectified effect of the variability on tracer fields harder to measure. On the other hand, it is instructive is substantial (Figs. 3c,d). Climatological maps of PV to see the substantial differences between instanta- fields show some qualitative similarities with our PV neous tracer (Figs. 2g–i), time-averaged tracer (Fig. 3c), fields (Figs. 2d–f) and some important differences, too and tracer from the time-averaged circulation (Fig. 3d). (Keffer 1985; Talley 1988). There are regions of homo- There is very little in common between them that em- geneous PV and also areas of uniform zonal PV con- phasizes the difficulty of interpreting a synoptic section tours, just as in our model solution, but the diversity of of transient tracer data. Apart from the problem of PV field in the real ocean is much greater. Similarly, the mapping a sparsely sampled transect, an instantaneous CFC-11 concentration age fields from the real ocean field should not be confused either with a time- have some resemblance to the ideal age structure seen averaged field, or with the tracer field resulting from in our model (Figs. 2g–i). In particular, the highest ages are along the eastern and southern boundaries of the the time-averaged circulation. subtropical gyre (see, e.g., Warner et al. 1996, their Fig. There exist interesting relationships between tracer 6; Fine et al. 2002). and potential vorticity. To be specific, there is high At best, our model flow is a crude caricature of the ideal age (and hence ⌫)/potential vorticity anticorrela- natural, midlatitude, middepth, ocean circulation, how- tion in the subtropical gyre and jet region, with low ever. Perhaps the most important shortcoming of the correlation elsewhere (with some nonrandom structure present model in comparison with the real ocean is the caused by basin modes). Almost identical correlations lack of vertical stratification. In our barotropic flow, PV exist between PV and the other tracer ages we studied, is forced throughout the domain by the wind stress curl, because of properties 1–4 (Fig. 4). The relationship can Sq. In the interior ocean away from the surface mixed be explained, a posteriori, by the different sources/sinks layer, PV sources vanish, however, and PV is materially of PV and tracer and the homogenizing effects of eddy- conserved on isopycnals. We have partly investigated flux arising from the unsteady flows. Knowledge of the this issue in Fig. 9c in an experiment with vanishing time-average ideal age and PV fields was presumed in wind stress in part of the domain. In this case we find the reasoning advanced in section 3d, however. We did the strong ideal age/PV anticorrelation is still present not anticipate the correlation structure in Fig. 8 and although the structure is modified. This finding suggests could not give an adequate explanation without knowl- that the existence of high correlation does not depend edge of the time-mean fields. much on the detailed PV dynamics. There is another From these results we speculate that regions of important difference between our model and the real strongly correlated tracer age and PV exist in the real ocean: Our deformation radius is comparable to the ocean. Our understanding of this relationship is not yet model domain size because there is only one layer, and mature enough to predict where these regions might be PV anomalies are therefore associated with relative and mapping them using infrequent synoptic sections vorticity, not layer thickness changes (Table 1). The will not be easy. Nevertheless, time series of ocean trac- real (baroclinic) deformation radius is small relative to ers are slowly becoming available and direct correla- the size of the ocean basins and large-scale PV anoma- tions may become feasible at a few locations. Where a lies are therefore associated with thickness changes. It reliable connection between PV and tracer age can be is unlikely that the results on tracer age relationships established, PV anomalies can be used as proxies for are sensitive to this issue, however, for the reasons tracer anomalies, as has been done in the stratosphere. stated above. Similarly, the existence of regions of In turn, tracer anomalies reflect variations in the TTD strong PV and tracer age covariability is probably ro- itself, as described above. Hence, observations of PV bust although the exact nature of that covariability does variability could be related to TTD changes. Given the depend on the circulation, tracer sources, and PV dy- current interest in observing how interior perturbations namics (section 3d).

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5. Conclusions and ⍀ is the part of the ocean surface where tracer ϭ ␦ Ϫ Ј Our results show evidence to support the following enters. The boundary condition for G is G (t t )on ⍀ conclusions. and no-flux elsewhere. The TTD constructs the interior tracer concentration C(r, t) from the surface tracer 1) Tracer ages and the mean transit time and width of concentration C(⍀, t)by the underlying transit-time distribution are accurately related through properties 1–4 in unsteady t C͑r, t͒ ϭ ͵ C͑⍀, tЈ͒G͑r, t|⍀, tЈ͒ dtЈ, ͑A2͒ flows (see section 3c). Tracers with exponential or Ϫϱ low-order polynomial source functions can be used to accurately estimate the mean transit time and where C(r, t) satisfies width of the TTD. Tracer age variations from these Ѩ tracers are very tightly related in phase and ampli- C ϩ L͑C͒ ϭ 0. ͑A3͒ tude. Ѩt 2) Temporal perturbations in potential vorticity and ideal age (hence mean transit time by property 1) Properties 2–4 may be seen as follows. are highly anticorrelated in some, but not all, areas of our model domain (see section 3d). We have a. Property 2 achieved a partial understanding of this relationship. Let tracer 1 be materially conserved with an expo- The existence of the relationship is insensitive to ⍀ ϭ ␭ nential source function. That is, C1( , t) exp( t) and, circulation, tracer sources, and PV dynamics, al- using (A2), we have though the details are. We speculate that strong anticorrelations between PV and tracer age exist in the t ͑ ͒ ϭ ͵ ␭tЈ ͑ |⍀ Ј͒ Ј ͑ ͒ real ocean, although we cannot say where. C1 r, t e G r, t , t dt . A4 Ϫϱ These conclusions imply that variations in integrated ϭ Ϫ␭ transport (i.e., changes in the TTD mean and width) Now define tracer 2 so that C2 exp( t)C1. Tracer 2 might be estimated using common oceanographic mea- therefore satisfies surements such as isopycnic spacing from hydrography ѨC (hence PV) and dissolved oxygen concentration. Con- 2 ϩ ͑ ͒ ϭϪ␭ ͑ ͒ Ѩ L C2 C2 A5 firmation for more realistic flow is now required. Our t future work also lies in two other directions. We seek to with a unit source on ⍀. Using the definitions of Table diagnose and understand the links between PV and the 2 we see that the tracer age of tracer 1, the exponential TTD itself for unsteady flows. We also aim to exploit tracer, is these findings to improve knowledge of middepth transport in the North Atlantic Ocean using real tracer t ␶ ͑ ͒ ϭ Ϫ ␭Ϫ1 ͵ ␭tЈ ͑ |⍀ Ј͒ Ј ͑ ͒ observations. 1 r, t t ln e G r, t , t dt , A6 Ϫϱ Acknowledgments. Baylor Fox-Kemper made several and the tracer age of tracer 2, the radioactive tracer, is helpful suggestions. This work was supported by grants from the physical oceanography program at NSF t ␶ ͑ ͒ ϭϪ␭Ϫ1 Ϫ␭t͵ ␭tЈ ͑ |⍀ Ј͒ Ј (OCE-9911318 and OCE-0326670). 2 r, t lne e G r, t , t dt Ϫϱ

APPENDIX t ϭ Ϫ ␭Ϫ1 ␭tЈ |⍀ Ј Ј ͑ ͒ Derivation of Tracer Age Relationships t ln͵ e G͑r, t , t ͒ dt , A7 Ϫϱ We demonstrate here that properties 2–4 of section ␶ 3c hold for unsteady flow, following Haine and Hall which is 1(r, t), and hence property 2 holds. (2002) and Waugh et al. (2003). The TTD, or boundary propagator G(r, t|⍀, tЈ), sat- b. Property 3 isfies The tracer age ␶(r, t) for an exponential tracer can be ѨG written as ϩ L͑G͒ ϭ 0, ͑A1͒ Ѩt ϱ Ϫ Ϫ␭ Ј where L is the advection-diffusion operator [generaliz- ␶͑r, t͒ ϭϪ␭ 1 ln͵ e tG͑r, t|⍀, t Ϫ tЈ͒ dtЈ, ͑A8͒ ing the operator in (6)], r is a point in the ocean interior, 0

Unauthenticated | Downloaded 09/25/21 03:30 PM UTC 2266 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 35 using (A6). The Taylor expansion of exp(Ϫ␭tЈ) about Namely, C(r, t) ϭ C[⍀, t Ϫ ␶ (r, t)] gives the following the mean transit time ⌫ gives polynomial for the tracer age:

ϱ ϱ ͑tЈ Ϫ ⌫͒n N n n ␶͑ ͒ ϭϪ␭Ϫ1 ͵ ͑Ϫ␭͒n ͑Ϫ ͒mͩ ͪ nϪm͑␶m Ϫ ͒ ϭ ͑ ͒ r, t ln ͚ ͚ an ͚ 1 t mm 0 A16 0 nϭ0 n! nϭ1 mϭ1 m

Ϫ␭⌫ ϫ e G͑r, t|⍀, t Ϫ tЈ͒ dtЈ (notice the lower limits of the sums are now 1 not zero, ␶0 Ϫ ϭ because m0 0 by normalization). This expres- ϱ ␮ n sion, which proves property 4, equals (15) of Waugh et ϭ ⌫ Ϫ ␭Ϫ1 ln͚ ͑Ϫ␭͒n nϭ0 n! al. (2003), although in that article we explicitly assumed steady flow. ϱ ␮ Note that for liner tracers (N ϭ 1) we simply have ϭ ⌫ Ϫ ␭Ϫ1 ϩ ␭2⌬2 ϩ Ϫ␭͒n n lnͫ1 ͚ ͑ ͬ, ␶ ϭ⌫ ϭ nϭ3 n! [from (A16)], while for quadratic tracers (N 2) the interior concentration is ͑A9͒ C ϭ a ϩ a ͑t Ϫ ⌫͒ ϩ a ͓͑t Ϫ ⌫͒2 ϩ 2⌬2͔ ͑A17͒ where the centered moments are 0 1 2 from (A15). These expressions are exploited in section ϱ ␮ ͑ ͒ ϭ ͵ ͑ Ј Ϫ ⌫͒n ͑ |⍀ Ϫ Ј͒ Ј ͑ ͒ 3c. Last, note that property 3 is consistent with property n r, t t G r, t , t t dt , A10 0 4 because cubic terms and higher have been neglected in (A9) to reach (A11). ␮ ϭ ␮ ϭ ␮ ϭ ⌬2 ␭ and 0 1, 1 0, and 2 2 . For small enough and |(n ϩ 1)␮ /␮ ϩ | Ͼ ␭ [i.e., successive moments do n n 1 REFERENCES not grow so fast that the sum in (A9) diverges] we expand the logarithm to give property 3: Allaart, M. A. F., H. Kelder, and L. C. Heijboer, 1993: On the relation between ozone and potential vorticity. Geophys. Res. ␶͑r, t͒ Ӎ ⌫ Ϫ ␭⌬2. ͑A11͒ Lett., 20, 811–814. Chang, K. I., M. Ghil, K. Ide, and C. A. Lai, 2001: Transition to c. Property 4 aperiodic variability in a wind-driven double-gyre circulation model. J. Phys. Oceanogr., 31, 1260–1286. Consider a tracer with a polynomial source function, Cox, M. D., 1985: An eddy resolving numerical model of the ven- N n tilated thermocline. J. Phys. Oceanogr., 15, 1312–1324. C(⍀, t) ϭ⌺ϭ a t . The interior concentration is n 0 n Danielsen, E. F., 1968: Stratospheric-tropospheric exchange based N t on radioactivity, ozone and potential vorticity. J. Atmos. Sci., ͑ ͒ ϭ ͵ Јn ͑ |⍀ Ј͒ Ј 25, 502–518. C r, t ͚ an t G r, t , t dt nϭ0 Ϫϱ Deleersnijder, E., J. Campin, and E. J. M. Delhez, 2001: The concept of age in marine modeling, Part I. Theory of prelimi- N ϱ nary model results. J. Mar. Syst., 28, 229–267. ϭ ͵ ͑ Ϫ Ј͒n ͑ |⍀ Ϫ Ј͒ Ј ͑ ͒ ͚ an t t G r, t , t t dt . A12 Delhez, E. J. M., E. Deleersnijder, A. Mouchet, and J.-M. Beck- nϭ0 0 ers, 2003: A note on the age of radioactive tracers. J. Mar. Now we write Syst., 38, 277–286. Doney, S. C., W. J. Jenkins, and J. L. Bullister, 1997: A compari- n n son of ocean tracer dating techniques on a meridional section ͑t Ϫ tЈ͒n ϭ ͚ ͑Ϫ1͒mͩ ͪt nϪmtЈm ͑A13͒ in the eastern North Atlantic. Deep-Sea Res., 44A, 603–626. mϭ0 m England, M. H., 1995: The age of water and ventilation timescales in a global ocean model. J. Phys. Oceanogr., 25, 2756–2777. from the binomial theorem, use the definition of the ——, and E. Maier-Reimer, 2001: Using chemical tracers to assess raw moments, ocean models. Rev. Geophys., 39, 29–70. Figueroa, H. A., 1994: Eddy resolution versus eddy diffusion in a ϱ ͑ ͒ ϭ ͵ Јn ͑ |⍀ Ϫ Ј͒ Ј ͑ ͒ double gyre GCM. Part II: Mixing of passive tracers. J. Phys. mn r, t t G r, t , t t dt , A14 Oceanogr., 24, 387–402. 0 Fine, R. A., M. Rhein, and C. Andrie, 2002: Using a CFC effective and see that age to estimate propagation and storage of climate anomalies in the deep western North Atlantic Ocean. Geophys. Res. N n n Lett., 29, 2227, doi:10.1029/2002GL015618. ͑ ͒ ϭ ͑Ϫ ͒mͩ ͪ nϪm ͑ ͒ C r, t ͚ an ͚ 1 t mm. A15 Ghil, M., Y. Feliks, and L. U. Sushama, 2002: Baroclinic and baro- nϭ0 mϭ0 m tropic aspects of the wind-driven ocean circulation. Physica D, 167, 1–35. The interior tracer concentration thus depends only on Haine, T. W. N., and S. L. Gray, 2001: Quantifying mesoscales the first N moments of the TTD. This statement must variability in ocean transient tracer fields. J. Geophys. Res., also therefore be true for the associated tracer age, ␶(r, t). 106, 13 861–13 878.

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