We apply to Classical Labrador Sea Water (CLSW) the transit-time distribution (TTD) method to estimate the inventory and uptake of anthropogenic carbon dioxide (Cant). A model of TTDs representing bulk-advection and diffusive mixing is constrained with CFC11 data. The constrained TTDs are used to propagate Cant into CLSW, allowing the air-sea disequilibrium to evolve consistently. Cant in the Labrador Sea (LS) surface waters cannot keep pace with increasing atmospheric CO2 and is highly undersaturated. Our best estimate for 2001 is an anthropogenic inventory of 1.0 Gt C and an uptake of 0.02 Gt C/year. By additionally using the constraint of present-day CO2 measurements, we estimate that the preindustrial LS was neutral or a weak source of CO2 to the atmosphere. Our estimates are subject to possible error due to the assumption of steady-state transport and carbon biochemistry.
 The North Atlantic (NA) is considered the ocean region with the largest storage of anthropogenic dissolved inorganic carbon (Cant). Large concentrations of Cant may be present at depths below 1.5 km [e.g., Sabine et al., 2004], the result of the formation and spreading of North Atlantic Deep Water, of which Classical Labrador Sea Water (CLSW) is a major component. However, considerable uncertainties remain in the NA and globally, in part due to approximations built into the methods of inference. Here we focus on the uptake of Cant by LS, where CLSW is formed in winter by deep convection in the cyclonic gyre of the LS [e.g., Khatiwala et al., 2002]. This weakly stratified water mass can reach depths greater than 2 km. To estimate LS uptake and inventory of Cant we apply the TTD technique [Hall et al., 2004], which avoids commonly made assumptions of weak-mixing and constant disequilibrium.
 The concentration of any passive inert tracer χ, averaged over an isopycnal volume V, can be expressed as a weighted sum over all its past outcrop values:
where ��V(τ) is the V-averaged TTD since water in V was last in contact with a formation region of sea-surface area A, and χA(t) is the concentration history averaged over the formation-region. By applying (1) to anthropogenic dissolved inorganic carbon, Cant, we exploit the widely-made assumptions that Cant penetrates the ocean as a passive and inert tracer along isopycnals, and the ocean circulation is in steady state [e.g., Sabine et al., 2004]. Although the error incurred by this assumption is of considerable interest, we do not address it here. Our analysis should be viewed as a steady-state projection of the transport information gleaned from modern-day tracers to Cant uptake over the industrial era.
 To estimate ��V we first apply (1) to CFC11, a passive, inert transient tracer with no natural background. Both CFC11V(t) and CFC11A(t) can be estimated from observations, allowing us to invert (1) for ��V(t). Because we assume steady state circulation, once ��V is estimated, it can be applied to Cant(t) at any time. We assume ��V(t) to have a domain-averaged “inverse Gaussian” functional form [Hall et al., 2004], where the two parameters are the mean residence time, τM, and Peclet number, Pe. The high Pe limit corresponds to bulk-advective transport (weak mixing), while the low Pe limit corresponds to diffusive transport (strong mixing). Pe and τM are allowed to range freely, subject to the CFC11 constraint applied via (1). No assumptions are made about the relative importance of bulk advective and diffusive transport, in contrast to other studies that assume mixing to be weak [e.g., Gruber, 1998; Sabine et al., 2004]. Since the single datum, CFC11V, cannot simultaneously constrain Pe and τM, a family of ��V is obtained. The ��V are applied to CantA(t), giving a range of values, from a minimum for strong mixing (low Pe) to a maximum for weak mixing (high Pe). However, the multiple-tracer analysis of Waugh et al.  showed the weak-mixing limit to be unrealistic. Here, results are quoted in the weak-mixing limit, and otherwise noted. In the figures the weak- and strong-mixing estimates are both shown, along with their individual uncertainties. As it turns out, LSW parameters are such that the difference between our carbon inferences in the weak- and strong- mixing limits is smaller than other uncertainties.
 We use chlorofluorocarbons data from the WOCE Campaign that have been objectively mapped onto the CLSW density range σθ = 27.897 − 27.985 (D. A. LeBel et al., The Distribution of CFC-11 in the North Atlantic During WOCE: Inventories and calculated water mass formation rates, submitted to Deep-Sea Research, Part I, hereinafter referred to as LeBel et al., submitted manuscript, 2006). CLSW CFC11 concentrations are shown in Figure 1. A time-averaged convective column in the LS (the formation region) is assumed to be the sole source for this water. To obtain the CFC11 inventory, ICFC, we integrate the gridded concentrations over the CLSW volume V from the northern edge of the formation region to 42°N. South of 42°N CLSW signal is weak, and there may be contamination of Mediterranean Outflow Water. We find V = 9 · 1015 m3 and ICFC = 22 · 106 moles.
 We estimate the CFC11 formation-region history by scaling the atmospheric history [Walker et al., 2000] to match the observed LS values, a procedure which implicitly allows for subsaturation of CFC11A(t). This is equivalent to applying a solubility function to atmospheric CFC11(t) and multiplying the resulting time series by a single saturation fraction. The CFC11 saturation implied by our scaling is 66%, consistent with 60% of Smethie and Fine .
 To evaluate CantA(t), it is often assumed that the anthropogenic contribution to the air-sea disequilibrium is small compared to the natural background [e.g., Gruber et al., 1996; Sabine et al., 2004]. This “constant disequilibrium” assumption allows for the use of equilibrium inorganic carbon chemistry to estimate CantA(t) from the partial pressure of anthropogenic CO2 in the atmosphere, given temperature, salinity, and alkalinity of the water mass under consideration [Thomas et al., 2001]. However, at least on a global scale the disequilibrium must have evolved over the industrial era, because preindustrially the atmosphere and the ocean were in approximate equilibrium, while in the present day the atmosphere leads the ocean in pCO2, due to anthropogenic emissions. To avoid the assumption of constant disequilibrium we follow the procedure of Hall et al. , who solve the following equation for CantA(t):
where pCO2,anta(t) is the partial pressure of CO2 in the LS that would be in equilibrium with the atmospheric CO2 and is calculated using CO2 solubility [Weiss and Price, 1980], pCO2,anto(t) is the actual anthropogenic CO2 partial pressure in the LS mixed layer, and Kex is the annual-mean air-sea exchange coefficient. (We use annual-mean values for Kex and A to be consistent with our calculation of ��V, which has no seasonality.) Equation (2) is the statement that the air-sea flux into the LS must equal the rate of change of the inventory within CLSW, expressed as (ρV · CantV), with CantV from equation (1). To solve expression (2) a relationship between pCO2,anto and CantA is required. The total quantities (preindustrial plus anthropogenic) pCO2o and CA are related by definition by the equilibrium inorganic carbon system, pCO2o = f(CA), and so the anthropogenic quantities are related as pCO2,anto(t) = f(CA(t)) − f(CA(1780)). (We use the equilibrium carbon chemistry coefficients of Dickson and Millero  and the empirical relationship for alkalinity, AT, in terms of salinity, S, described by Millero et al. , AT=51.24·S +520.1.) Apparently, the unknown preindustrial state must be specified. However, given a modern-day observations of CA at a date tobs (e.g., the 1995 climatology of Takahashi et al. ), we have CA(1780) = CA(tobs) − CantA(tobs). Therefore,
Substitution of (3) into (2) results in a nonlinear integral equation in the single unknown CantA(t), which we discretize and write in matrix form. The resulting nonlinear system of equations is solved using Newton's method.
3.1. Anthropogenic Carbon
Figure 2 shows pCO2,anta(t) and our best estimates of pCO2,anto(t) in the weak- and strong-mixing limits. Also shown is the impact of uncertainty in Kex and A (see below). The anthropogenic signal in the LS mixed layer is 17–38% and 15–35% the concurrent atmospheric value in the strong- and weak-mixing limits. Once pCO2,anto is obtained we can evaluate the anthropogenic inventory, Iant, using equation (1) and the flux (uptake) Fant = d Iant/dt. Figure 3 shows our best estimates of Iant and Fant in the weak- and strong-mixing limits. The inventory and uptake have increased steadily over the industrial era, driven in our analysis solely by changes in pCO2,anta. (In reality, other factors likely play a role, such as variation in ocean transport and carbon biochemistry.) In 2001 we estimate in the low Peclet limit an inventory 1.0−0.4+0.3 Gt and an uptake 1.8−0.8+0.5 × 10−2 Gt/yr, where the ranges include uncertainties in Kex, A, and pCO2(1995) (see below).
 Uncertainty in Kex and A significantly impact our anthropogenic carbon estimates. Our best estimate, Kexbest, is the annual average of the North Atlantic values reported by Carr et al. , based on QuickSCAT winds, assuming a velocity-squared dependence. Carr et al.  report an 8% uncertainty in wind speed, but our effective error is larger, because we use only a fraction of one of their area-averaged Kex. Moreover, because we use only annual averages, any covariance between Kex and CLSW formation is neglected. Finally, there remains uncertainty as to the nature of the wind-speed dependence of Kex. Our best estimate surface area, Abest, is a 500 km × 600 km region, centered in the LS (LeBel et al., submitted manuscript, 2006). However, CLSW formation is known to be episodic in intensity, location, and areal extent. To estimate the impact of these uncertainties we sample randomly from the uniform distributions Kex: 0.5Kexbest → 1.5Kexbest and A: 0.5Abest → 2Abest, for each pair computing pCO2,anto in the weak to strong mixing limits. For pCO2o(1995) we use the work of Takahashi et al. , and we assign an uncertainty range of ±10 ppm, from which we also sample randomly. We note that LS pCO2 measurements reported by DeGrandpre et al.  are in reasonable agreement with Takahashi's climatology. The value of pCO2o(1995) has negligible impact on the anthropogenic estimates, because carbon chemistry is only weakly nonlinear in the range of anthropogenic CO2 variation. (In the limit of strictly linear chemistry, there is no dependence of the anthropogenic perturbation on the preindustrial state, and pCO2o(1995) is not required.) The shaded region in Figure 2 represents one standard deviation above and below the mean computed over a number of such calculations, in the case of each the weak-mixing and strong-mixing limits.
3.2. Preindustrial Carbon
 Once pCO2,anto is known, we compute the total outcrop partial pressure as pCO2o(t) = pCO2o(1780) + pCO2,anto(t). As mentioned in Section 2, pCO2o(1780) is determined by the requirement that pCO2o(1995) matches the 1995 annually-averaged observationally-based estimate of Takahashi et al. . Figure 4 shows the resulting best estimate pCO2o. Our best estimate is pCO2o(1780) = 285 ppm, which is higher than the atmospheric value in 1780 of 278 ppm, suggesting that in 1780 the LS was a source of CO2 to the atmosphere. It should be noted that zero flux is encompassed by the uncertainty around our best estimates. Moreover, we have not estimated the error of neglecting temporal variation in transport and biochemistry, which may be significant. We therefore do not assign a sign to the preindustrial flux. The uptake history is shown in Figure 5. Sometime between 1800 and 1960 pCO2o became significantly larger than pCO2a, after which the LS became a clear sink of atmospheric CO2.
 We find that pCO2,anto in LS surface waters cannot keep pace with rapidly increasing atmospheric pCO2,anta and in 2001 was only 17%–38% of the concurrent atmospheric value in the strong-mixing limit. Constant air-sea disequilibrium is a poor approximation. The corresponding 2001 anthropogenic inventory in CLSW is 1.0−0.4+0.3 Gt, and the uptake 1.8−0.8+0.5 × 10−2 Gt/yr. The small anthropogenic signal, when combined with present-day observations of total pCO2o, implies to a preindustrial pCO2o slightly higher than the preindustrial atmospheric value, suggesting that preindustrial LS was a weak source of carbon to the atmosphere, but this is not significantly different than zero, given uncertainties. Interestingly, in simulations by the MIT GCM with carbon biochemistry the preindustrial LS is also a source of CO2 to the atmosphere, at least in winter (M. Follows, personal communication, 2005). The mechanisms for this outgassing are unclear and warrant further investigation. The convective plumes may entrain, then mix upward, deep water rich in dissolved inorganic carbon.
 In contrast, Hall et al.  found pCO2,anto in Indian Ocean surface waters to track closely pCO2,anta. The major parameter difference between the two analyses is the length scale d = V/A (the ratio of the domain volume to the outcrop area), which for the LS is roughly 100 times larger than for the Indian Ocean thermocline. Consider the weak-mixing limit. ��V takes the form GV(t) = τadv−1 for t < τadv, and zero thereafter, where τadv is τM in the weak-mixing limit. For simplicity, we assume linear carbon chemistry: Cant = γpCO2,anto. (In fact, Fant(2001) inferred using full nonlinear carbon chemistry and chemistry linearized about an intermediate pCO2,anta differ by only 0.2%.) Equation (2) becomes:
(for t − 1780 > τadv), where UP = Kex/(ργ) is the “piston velocity” at which material crosses the air-sea interface, and U = d/τadv is the speed at which material gets transported into the interior. There are two limiting cases: For UP ≫ U material builds up very quickly in surface waters compared to the rate at which it is transported away. In this limit the atmosphere and ocean are able to equilibrate, i.e., pCO2,anto ≈ pCO2,anta. This turns out to be the relevant limit in the Indian Ocean. By contrast, the LS is closer to the other limit: UP ≪ U. The transport across the sea surface cannot keep pace with the transport away from the surface. Consequently pCO2,anto ≪ pCO2,anta.
 We have assumed that the air-sea exchange relevant for CLSW occurs predominantly in the LS, although we have allowed for considerable uncertainty in sea-surface area. It is possible, however, that during the roughly one year required for CO2 to equilibrate across the air-sea interface, Cant from surface regions outside the LS is transported into the LS and supplies CLSW. The area, A, over which the relevant CLSW air-sea exchange occurs is then effectively larger. To estimate roughly the impact of this effect assume that the CLSW formation rate is 5 Sv [e.g., Smethie and Fine, 2001], and that this formation is supplied by a slab 200 m thick converging on the LS. In one year continuity demands that the slab shrink by an area 7.8 × 105 km2. This effective area of air-sea exchange is about 2.6 times larger than our best estimate, and it impacts our Cant calculations as follows (in the low Pe limit): the present-day surface saturation is 40%, the uptake is 0.04 Gt C/y, the inventory is 1.9 Gt C, and the preindustrial LS is 278 ppm.
 Finally, recent estimates of trends in subpolar North Atlantic pCO2 in surface waters based on direct measurements indicate a rate of increase greater than the atmospheric trend [Omar and Olsen, 2006; Léfevre et al., 2004], opposite our CLSW result. This is not contradictory, because these studies have not focused on CLSW, which may behave differently. Indeed, Tanhua et al.  find CLSW to be the one North Atlantic water mass whose Cant uptake over the period 1981–2004 is significantly less than what would be expected if surface waters kept pace with the atmosphere.
 This work has been supported by NSF grant OCE-0326860, NASA, and a NSF IGERT fellowship to FT at Columbia University. SK was funded by NSF grant ATM-0233853. We thank M. Holzer, F. Primeau, and D. Waugh for comments. We are grateful for the efforts of the many researchers involved in making the CFC measurements. This is LDEO contribution 7013.