The use of CFC-11, CFC-12, and SF6 to quantify oceanic ventilation rates, interior water age, and formation rates requires knowledge of the saturation levels at the sea surface. While their atmospheric histories are relatively well known, physical processes in the mixed layer in conjunction with limited air-sea gas exchange can cause surface concentrations to be in disequilibrium with the atmosphere. We use an offline tracer advection-diffusion code that evolves tracers using along-isopycnal and cross-isopycnal mass fluxes from a global, climatological run of the Hallberg Isopycnal Model to reconstruct the saturation level of all three tracers over the entirety of their atmospheric histories. Disequilibria on a global scale occur in regions associated with deep winter mixed layers and are found throughout the time period of the release of these chemicals into the atmosphere. Sensitivity studies using targeted model simulations, focusing on the North Pacific, show that seasonal cycles in temperature and salinity that affect gas solubility as well as entrainment of water containing low concentration of tracers during mixed layer deepening are the dominant causes of undersaturation. When using the transit time distribution method, our results show that these undersaturations introduce a significant bias toward older ages for North Pacific Central Mode Water but do not significantly affect estimates of anthropogenic carbon inventory.
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 Transient, anthropogenic gases such as the chlorofluorocarbons CFC-11 and CFC-12 have been used extensively as tracers of oceanic ventilation. These compounds have been released in detectable amounts since the 1940s and have atmospheric mixing ratios whose time history can be estimated well from observations or inferred from anthropogenic activity [Walker et al., 2000] (Figure 1a). Due to measures undertaken as a result of the Montreal Protocol in 1987, atmospheric CFC concentrations have begun to decrease. The complications that arise from this nonmonotonic history and from the similarity between the CFC-11 and CFC-12 source functions [Sonnerup, 2001] has spurred the extension of tracer techniques to include sulfur hexafluoride (SF6), another anthropogenic gas which has been released in a exponentially increasing manner since the mid-1950s [Bullister et al., 2006; Sonnerup et al., 2008].
 One application of transient tracers, the transit time distribution (TTD) method, described by Haine and Hall  has provided a framework to estimate the mean age and width of the distribution that describes the spectrum of the ages in a water parcel by inferring the advective-diffusive Green's function from in situ tracer observations. This problem is highly underconstrained, but the use of CFCs and SF6 in tandem can give better estimates of the TTD [Hall et al., 2002; Waugh et al., 2002]. As noted by Hall et al. , a significant source of error in these calculations is the disequilibrium in tracer concentration between the atmosphere and oceanic mixed layer. The most important aspect of this disequilibrium is gas undersaturation with respect to the atmosphere (i.e., negative disequilibrium) observed at the end of winter when permanent subduction into the thermocline occurs [Cushman-Roisin, 1987] or during water mass formation through deep convection [Marshall and Schott, 1999]. Undersaturations during water mass subduction/formation can significantly skew estimates of ventilation ages and pathways from tracers as well as oceanic anthropogenic carbon concentration using the tracer-based TTD [Waugh et al., 2004; Tanhua et al., 2009] or ΔC* methods [Gruber et al., 1996].
 Studies utilizing CFC and SF6 concentrations to infer ages or TTDs have employed a variety of methods to address the biases introduced by undersaturation. Most researchers assume a constant fixed value of saturation over time [e.g., Waugh et al., 2004; Tanhua et al., 2009]. For example in a global study, Khatiwala et al.  used mixed layer undersaturations observed during the World Ocean Circulation Experiment (WOCE) era to scale the CFC mixed layer boundary condition derived from atmospheric concentrations. Using a one-dimensional slab mixed layer model, Haine and Richards  showed a seasonal variability of 70–100% saturation for CFC-11 and CFC-12. Mecking et al. , by constraining a 2-D advection-diffusion model to reproduce observed interior concentrations, estimated the CFC saturations at isopycnal outcrops in the North Pacific to be as low as 80%. Local one-dimensional mixed layer models tuned to the region of North Pacific Central Mode Water Formation in the 1990s gave similar low saturation values for the North Pacific [Tokieda et al., 2004; Mecking et al., 2006] while CFC saturation in regions of seasonal ice coverage, such as the Antarctic margins, may be even lower [Rodehacke et al., 2010].
 In this paper, we use an offline tracer transport code described in section 2 to estimate the temporal and spatial variability in the surface saturation of CFC-11, CFC-12, and SF6. We define undersaturation (λ) by
which describes the percent deviation from equilibrium when the observed or modeled tracer concentration (Cobs) is smaller than the saturation value (Csat). By this definition, negative values of λ represent oversaturation, and 100 − λ is equivalent to percent saturation.
 Model results are presented in section 3. We show global maps of undersaturation to identify spatial patterns (section 3.1) as well as time series of undersaturation at a fixed location (section 3.2). We then evaluate the role of physical processes controlling undersaturation in the North Pacific by performing a series of model experiments to isolate the effects of the seasonal cycle in saturation concentration, gas transfer velocities, and wintertime entrainment of subsurface waters (section 3.3). The runs performed for this paper are summarized in Table 1. The discussion of these results and conclusions are presented in sections 4 and 5, respectively.
Monthly varying temperature, salinity, mass transport, and gas transfer velocity (scaled to global average of 15.6 cm h−1)b
Annually averaged gas transfer velocity for gas exchange
Gas transfer velocity (scaled to global average of 21.9 cm h−1)c
Annually averaged SST/SSS used to calculate solubility
Mixed layer entrainment does not change tracer concentration
Both modifications from Experiments 4 and 5 applied
Surface tracer concentrations set to saturation
2. Model Descriptions
 The research conducted herein relies on a two-tiered modeling approach. First, monthly averages of physical fields are derived from the Hallberg Isopycnal Model (HIM), a general circulation model solving the oceanic primitive equations of motion with realistic forcing. These fields are then used to drive an offline tracer advection-diffusion code with along-isopycnal mixing, dubbed Offtrac.
2.1. Hallberg Isopycnal Model
 Our run of HIM is similar to the default publicly available code distributed by NOAA's Geophysical Fluid Dynamics Laboratory (http://www.gfdl.noaa.gov/ocean-model) [Hallberg, 1995]. The horizontal grid is 210 cells in the meridional coordinate (nominally 1/3° resolution near the equator and 1° elsewhere) and 360 cells (1° resolution) in the zonal. The vertical resolution is determined by a distribution of 50 density interfaces (49 layers) referenced to 2000 db ranging from σ2 = 21.85–37.90 kg m−3. The surface ocean is a Kraus-Turner bulk mixed layer with Richardson-based turbulence [Kraus and Turner, 1967] and two buffer layers allowing smooth detrainment/entrainment to outcropping isopycnal surfaces. Transport due to unresolved eddies is parameterized as along-isopycnal diffusion, an isopycnal thickness diffusion equivalent to a Gent-McWilliams scheme [Gent and McWilliams, 1990] (each with a diffusivity of 600 m2 s−1), and cross-isopycnal diffusion with a diffusivity of 10−5 m2 s−1.
 The model is initialized as a quiescent ocean with isopycnal depths, temperature, and salinity from the Steele et al.  climatology. Surface salinity is restored to climatology with a transfer velocity of 50 m/day by adding an implied freshwater flux. The model is spunup for 650 years using the normal year climatological Common Ocean Reference Experiment version 2 (CORE-2) forcing [Large and Yeager, 2009]. Previous applications of HIM include participation in the Coordinated Ocean-ice Reference Experiments Model Intercomparison [Griffies et al., 2009], North Pacific mode water studies [Ladd and Thompson, 2000, 2002] as well as investigation of oxygen variability in the North Pacific (using the Offtrac code) [Deutsch et al., 2005, 2006].
 Because CFCs and SF6 invade the ocean through air-sea gas exchange at the surface (except for a few SF6 tracer release experiments, with SF6 now being replaced by SF5CF3) [e.g., Ho et al., 2008], mixed layer processes play a crucial role in controlling the saturation states of CFCs and SF6. As a metric of how well these dynamics are represented in HIM, we compare the maximum mixed layer depths from the model (Figure 2a) to a mixed layer product based on Argo profiles by Holte et al.  (Figure 2b). The root-mean-square difference between the model and the observations amounts to 45 m globally and 30 m for the North Pacific.
 Visual comparison between Figures 2a and 2b shows that with the exception of the Brazil Current extension, the large-scale structure of maximum mixed layer depth in the model agrees well with observations, specifically in the South Indian, South Pacific, North Atlantic, and North Pacific Basins. In the western boundary current extensions, the regions of maximum mixed layer depth are larger in extent in the model than observed. This bias is typical of non-eddy resolving models because horizontal resolution is insufficient to fully resolve western boundary current dynamics [Smith et al., 2000; Penduff et al., 2010] or to adequately represent the eddy-controlled Antarctic Circumpolar Current [Hallberg and Gnanadesikan, 2006]. The modeled mixed layers also tend to be somewhat too deep because not all the processes controlling restratification are captured. For example, the western boundary current extensions are weaker than observed and do not advect warm water sufficiently quickly into the interior ocean, and also the restratification owing to the effect of mixed layer eddies is not adequately represented [Hallberg, 2003]. This tendency toward deeper mixed layers slows the time required for air-sea gas equilibration and accordingly may bias modeled undersaturations high. The Kuroshio and Kuroshio Extension (K/KE region) region off Japan, the subject of our more detailed analyses (see sections 3.2 and 3.3), is relatively well represented with maximum mixed layer depths being only about 10% deeper in the model and covering roughly the same geographic area as in the observations.
 To study the processes controlling surface disequilibria, particularly undersaturations of CFC-11, CFC-12, and SF6, we use the three-dimensional monthly averaged mass transports, surface wind speeds, temperature (T), and salinity (S) as well as monthly instantaneous isopycnal thickness fields from HIM to drive the offline model, Offtrac. These input fields to Offtrac, more specifically, are based on a normal year 12 month climatology created by averaging the monthly physical fields over the last 20 years of the 650 year HIM run. Tracer uptake was modeled using the one-dimensional air-sea gas exchange equation at each grid point:
where C is the concentration of the tracer, kw is the gas transfer velocity scaled by the Schmidt number of the gas [Wanninkhof, 1992; Zheng et al., 1998], H is the mixed layer depth, and Csat is the saturation concentration which for CFCs and SF6 is based on the tracers' empirically determined solubility functions [Warner and Weiss, 1985; Bullister et al., 2002] and atmospheric pressure. In Offtrac, gas exchange fluxes are calculated after advecting and diffusing tracers by integrating equation (2) using an Euler forward method with a roughly one day time step. Following the Ocean Carbon Modeling Intercomparison Project Phase 2 (OCMIP-2) guidelines [Najjar and Orr, 1998], the monthly averaged gas transfer velocities values were calculated by assuming a quadratic dependence on wind speed (U) [Wanninkhof, 1992]: kw = αU2. For consistency with the physical model, the U used for Offtrac is based on the CORE-2 forcing instead of the 1987–1992 SSM/I winds used in OCMIP [Najjar and Orr, 1998]. Following the assumptions of Sweeney et al. , we determined α by scaling kw to an average CO2 gas transfer velocity of 15.6 cm hr−1 (C. Sweeney, unpublished data, 2012). Through this scaling, bubble processes that can enhance gas exchange [Keeling, 1993; Schudlich and Emerson, 1996] but that are not explicitly parameterized in the model are accounted for to first order. The model configuration described here will be referred to as the control run (Experiment 1, Table 1) in the sections that follow.
 We primarily discuss the results from CFC-11 as being representative of the spatial patterns of mixed layer saturation levels for all three gases, but also examine differences between CFC-11, CFC-12, and SF6 when looking at the temporal evolution of undersaturation (section 3.2). For the latter and for the sensitivity experiments in section 3.3, we focus the analysis on the North Pacific because its ventilation dynamics, lacking deepwater formation, are simple relative to other basins and because mode water formation and corresponding mixed layer depth patterns in the North Pacific have been well studied with HIM [Ladd and Thompson, 2000-2002]. Additional figures, including global maps of CFC-12 and SF6 mixed layer saturations, are available in the supplemental material online (http://www.apl.washington.edu/project/project.php?id=cfc_mixed_layer). The model's sea surface temperature (SST), sea surface salinity (SSS), and mixed layer depth, along with the tracers' time-evolving surface concentrations and saturation levels are also made available online as netCDF files.
3.1. Regional Descriptions of Undersaturation
 The spatial structure of the saturation is relatively independent of time when climatological circulation fields are used to evolve the tracer structure. Hence, we first focus on the spatial pattern of maximum undersaturation of CFC-11 in 1980 (Figure 3) in combination with mixed layer depth (Figure 2), to show the boundary condition for relatively young (10–20 years), near surface waters measured during the 1990s WOCE era.
 In the open North Pacific, the largest undersaturations occur in the K/KE region off Japan (Figures 3a and 3b). This region is associated with deep mixed layers (yellow colors in Figure 2a) and the formation of North Pacific Subtropical and Central Mode Waters (the latter abbreviated as NPCMW) [Nakamura, 1996]. Additionally, large undersaturations can be seen in the Sea of Okhotsk, the source area for the waters that eventually form North Pacific Intermediate Water, and other marginal seas although the model resolves the dynamics in these areas less well. At the equator, strong upwelling in the east Pacific basin exposes older, low tracer concentration water to the surface which causes maximum undersaturations on the order of 5% there (Figure 3a). Similarly, coastal upwelling produces tracer undersaturations along the eastern margins of the Pacific and Atlantic Oceans.
 In the southern hemisphere, mode water formation sites of the Indian, Pacific, and Atlantic basins correspond to some of the deepest mixed layer depths of the world's oceans (yellow and red colors in Figure 2) and are associated with some of the largest undersaturations (as large as to 20%; Figure 3a). The patch of undersaturation east of the Antarctic Peninsula and north of the Weddell Sea is due to large amounts of winter ice coverage (thick, black contour line in Figure 3a) which isolates the surface ocean from the atmosphere preventing gas exchange.
 The North Atlantic basin shows a large coherent patch of undersaturation that stretches northeastward from the U.S. east coast and whose overall shape is consistent with the formation site of Eighteen Degree Water [Talley and Raymer, 1982], northeastward advection in the North Atlantic Current, and the formation of Labrador Sea Water [Talley and McCartney, 1982] which occurs eastward in the model off the tip of Greenland and not in the Labrador Sea.
 The sites of large maximum CFC-11 undersaturations (Figure 3a) that mostly occur in mid to late winter, correlate spatially with large maximal mixed layer depth (Figure 2a; R = 0.65 between maximum mixed layer and maximum CFC-11 undersaturation maps), also usually present at the end of winter and a prerequisite for mode water formation [McCartney, 1982]. Because of the frequent collocation of sites where large wintertime undersaturations occur with mode water formation sites (see above), we focus further analysis (sections 3.2 and 3.3) on the physical processes associated with those sites. Wintertime cooling results in buoyancy loss and weaker stratification which combined with momentum input from strong winds causes mixed layer deepening. The solubilities of CFCs and SF6 are temperature dependent so this cooling also increases their saturation concentrations driving a larger deviation of C from Csat (equation (2)). Deeper mixed layers also have a twofold effect leading to larger undersaturations: the tracer equilibration time scale is inversely proportional to H (equation (2)) and mixed layer deepening entrains older waters from below with relatively low tracer concentration. However, kw increases with wind speed so the higher winds may partially balance these disequilibria-causing effects (see section 3.3 for further analysis of these effects).
3.2. Temporal Evolution of Undersaturation
 To demonstrate how the degree of undersaturation varies over time, we examined the maximum undersaturation of CFC-11, CFC-12, and SF6 (Figure 4a) over a patch in the K/KE region where maximum CFC-11 undersaturation in 1980 was greater than a threshold value chosen at 8% here (black contour line in Figure 3b) and where North Pacific mode waters are formed (see section 3.1).
 The degree of maximum undersaturation declines over time as the ocean tracer concentrations become closer to equilibrium with the atmosphere (Figure 4a). Deviations from a smooth decrease are caused by changes in the growth rate of the atmospheric concentration of the tracers (Figure 1b). Generally, maximum undersaturations decrease faster during times of small atmospheric growth than during times of large atmospheric growth. For example, the CFC-11 atmospheric concentration increased relatively slowly from 1950 to 1960 (blue line in Figure 1b) such that the maximum CFC-11 undersaturation decreased from 19 to 13% (blue line in Figure 4a). From 1960 to 1975, the atmospheric growth accelerated while CFC-11 maximum undersaturation decreased very slowly from 12 to 11%. From 1975 to 1990, a fairly constant atmospheric CFC-11 growth rate caused a decrease in the undersaturation from 11 to 8%. The deceleration in the CFC-11 growth rate from 1990 to present day, including a decrease in atmospheric CFC-11 concentrations since 1994, allowed for a decrease in wintertime undersaturations to as low as ∼4% in recent years. The time series of CFC-12 undersaturations (red line in Figure 4a) is similar to that of CFC-11 with a less steep decline in maximum undersaturation (from 14 to 12%) in the initial part of the record (1950–1960). In contrast to CFCs, SF6 atmospheric concentrations (black lines in Figure 1) have grown monotonically at a roughly exponential rate over the course of its record. Accordingly, SF6's maximum undersaturation declines slowly overall, after the initial part of its record, from ∼10% in 1960 to its 2010 value of ∼7% (black line in Figure 4a). CFC undersaturations of 4–5% persist until present day (Figure 4a) likely due to the seasonal cycle of solubility (see section 3.3) and despite the slowing of the CFC atmospheric growth rates/reduction of atmospheric concentrations since the 1990s.
 As illustrated for CFC-11 in Figure 4b, tracer undersaturations exhibit a strong seasonal cycle imposed on the interannual behavior described above with the largest undersaturations occurring in late winter. Conversely in the summertime, the surface ocean is in near equilibrium due to the lower solubility of gases in higher temperature waters and a shallow mixed layer. Supersaturations may occur then if gas exchange cannot keep pace with the warming despite the shallower mixed layers. In the model, CFC-11 in the western part of the North Pacific does not become supersaturated beyond 1% (Figure 4b), whereas the eastern part does exhibit supersaturations as high as 5% in 1980 and beyond (not shown). This is, however, significantly less than what has been observed on some summer cruises [Warner, 1988]. The lesser supersaturations in the model may be attributed to the monthly advection-diffusion time step, masking the quick restratification and rapid heating of the mixed layer that can occur in the summer. Additionally, interannual variability in wind speed, particularly anomalously quiescent periods (not captured by the model because it is based on climatological forcing) may play a large role in allowing larger supersaturations in the observations.
 Focusing on the seasonal cycle for one model grid point (39.5°N, 163.5°E; see Figure 3b for location of grid point) illustrates that the month of the largest undersaturation at any point in the model does not necessarily correspond to the month of deepest mixed layers (dashed lines in Figure 5). Instead, it usually corresponds to the month which has the largest increase in mixed layer depth (Figure 5a) and/or largest decrease in temperature and hence increase in solubility (Figure 5b) which often occurs 1–2 months before mixed layer depths reach their maximum. SF6 solubility is less sensitive to temperature than the CFC solubilities. Accordingly, the maximum SF6 undersaturation is smaller in the early part of the SF6 record (1955–1975) compared to CFCs (Figure 4a). In 1980, at the model grid point shown in Figure 5, CFC-11, CFC-12, and SF6 all have comparable seasonal amplitudes in saturation state (Figure 5c), whereas in more recent years (using 2005 as an example, Figure 5d) SF6 is more undersaturated than CFCs by about 1–2% because of the slowing of the atmospheric CFC growth curves.
3.3. Sensitivity Experiments
 We perform a set of model experiments (Experiments 2–6, Table 1) to try to determine the contribution of gas exchange rates, seasonality of solubility based on T and S, mixed layer entrainment of older, lower concentration water in the winter, and the combined effect of the seasonal cycle and entrainment on the modeled tracer saturations (Figures 6 and 7). For reasons mentioned previously, we focus on the North Pacific region in the model and compare the resulting CFC-11 undersaturations in the sensitivity runs to the control run (Figure 3b; Experiment 1 in Table 1).
3.3.1. Dependence on Gas Transfer Velocities
 We performed two experiments to examine the sensitivity of the saturation level to gas transfer velocities. In Experiment 2, we replace the monthly varying kw with each grid point's annual average to evaluate the importance of seasonally varying wind speeds (Figure 6a). In response, the magnitude of maximum CFC-11 undersaturation in 1980 increases across the basin particularly in the K/KE region where undersaturations in the control run (Figure 3b) were already large. This suggests that high winter wind speeds (as present in the control run) do indeed act to reduce the wintertime disequilibrium in the mixed layer due to higher gas exchange rates (see section 3.1). In Experiment 3, instead of scaling the globally averaged kw (equation (2)) to the revised radiocarbon-based CO2 exchange rate of 15.6 cm h−1 (C. Sweeney, unpublished data, 2012), we scaled it to the higher, earlier estimate of 21.7 cm h−1 [Broecker et al., 1986]. The level of undersaturation across the North Pacific basin is dramatically reduced in this case (Figure 6b) compared to the control run (Figure 3b) with maximum CFC-11 undersaturation values in the K/KE region reaching only 8% in 1980. This suggests that the saturations reported in this paper are sensitive to the magnitude of kw and that model studies like OCMIP-2, that use the larger, globally averaged [Broecker et al., 1986] kw to scale to, might be underestimating the degree of disequilibria that CFCs and other gases experience in the mixed layer.
3.3.2. Sensitivity to Seasonality of Solubility
 In Experiment 4, to determine the effect of the seasonal cycle in SST and SSS (and accordingly solubility), we calculated Csat (equation (2)) in the mixed layer using each grid point's annually averaged temperature and salinity (Figure 7a) instead of using the monthly varying fields as in the control run (Figure 3b). In the southern and eastern parts of the North Pacific basin, the surface ocean remains essentially in equilibrium with the atmosphere in this case with maximum undersaturations somewhat closer to 0% than in the control run (Figure 3b). Tracer concentrations in the K/KE region and the Sea of Okhotsk are also much closer to equilibrium with the percent undersaturation being reduced to about half of its value from the control run. The basin-wide decrease in undersaturation suggests that the seasonal cycle in solubility and accordingly Csat (which are highest in winter when SST's are the lowest) is a significant contributor to the degree of undersaturation observed in the control run.
3.3.3. Sensitivity to Entrainment of Subsurface Water
 In Experiment 5 to remove the effect of the entrainment of older waters containing mostly lower (until recently) tracer concentrations, we do not allow tracer concentration to change when the mixed layer deepens and entrains water from below, although the tracer is still allowed to be transported horizontally. Thus, the mixed layer is entraining water with the same concentration as the mixed layer. In contrast to Experiment 4 (Figure 7a), the southern and eastern parts of the basin remain undersaturated to approximately the same (small) level (Figure 7b) as found in the control run (Figure 3b). The most dramatic difference in undersaturation between this experiment (Figure 7b) and the control simulation (Figure 3b) occurs in the K/KE region and the eastern edge of the Sea of Okhotsk. These waters are even closer to equilibrium (λ ≈ 5%) than in Experiment 4 (Figure 7a) indicating that entrainment of subsurface, tracer-poor water also plays an important role in the K/KE region.
3.3.4. Sensitivity to Both Seasonal Solubility Cycles and Entrainment
 In Experiment 6, we remove both the seasonal cycle in solubility (Experiment 4) as well as the effects of entrainment of subsurface water (Experiment 5). The removal of these effects in tandem yields surface waters that are essentially at equilibrium (λ < 2% everywhere in 1960 and beyond) for CFC-11 (Figure 7c) as well as for CFC-12 and SF6 (not shown). From this, we conclude that the seasonal cycle in solubility and entrainment of lower tracer waters are the main reasons for CFC and SF6 undersaturation in the mixed layer during those times. CFC-11 undersaturations larger than 2% existed before 1960, likely attributable to the relatively rapid growth of CFC-11 atmospheric concentrations then.
 Having identified two major processes that cause undersaturation (seasonal solubility cycles and entrainment), we examine differences in the temporal evolution of maximum undersaturation, averaged over the K/KE region (Figure 3b), between the control simulation and the experiments that isolate these effects (Experiments 4 and 5; Figure 8). Since 2000, CFC-11 is near saturation in Experiment 4 (blue line in Figure 8), while Experiment 5 (red line in Figure 8) shows undersaturations similar to those in the control run (black line in Figure 8). This suggests that in recent years entrainment has played only a minimal role in causing CFC-11 disequilibria because waters in the seasonal pycnocline are no longer tracer-poor relative to an atmosphere where CFC concentrations have stalled and declined. However, until 2000 (especially before ∼1980), the maximum CFC-11 undersaturation found in Experiment 5 is significantly less than in the control run, suggesting that entrainment was the dominant cause of disequilibria in the earlier part of the record. The effects of the seasonal solubility cycle have been relatively constant over time causing a 3–4% decrease in maximum CFC-11 undersaturation in the K/KE region compared to the control run throughout the record (see offset between blue and black lines in Figure 8).
 The model simulations shown here demonstrate that the saturation levels of trace gases vary greatly both in space and in time. To quantify the impact of undersaturation on tracer-inferred quantities, we discuss here an additional model run (Experiment 7) where the mixed layer concentrations were always set to Csat and hence are in equilibrium with the atmosphere. We find that the global CFC-11 inventory in this experiment yields an inventory that is 11% greater in 1970 than in the control run (Experiment 1) where dynamic gas exchange is included (Table 1). In 2000, this difference in inventory is only 7% due to the surface ocean being closer to equilibrium in recent years. Similarly, global inventories of CFC-12 and SF6 are 9.5% and 12.6% higher in 1970 and 6.4% and 7.4% higher in 2000, respectively, compared to the control run. Estimates of water mass formation rates that are based on CFC inventories [Smethie and Fine, 2001] or SF6 inventories will be affected accordingly since the formation rate is assumed to be proportional to the inventory.
 Mixed layer disequilibria can also cause biases in inferred ages of water parcels. Here, we examine the ages of water parcels using the 1-D TTD method [Haine and Hall, 2002; Waugh et al., 2003] which assumes that the TTD can be represented as an 1-D Inverse Gaussian determined by two parameters, a mean age Γ and a width parameter Δ. The ratio Γ2/Δ2 is equivalent here to the Peclet number (Pe) of the flow [Waugh et al., 2003]. In the following analysis, we convert tracer concentrations to an equivalent partial pressure assuming equilibrium with the atmosphere when interior waters left the surface ocean. We then infer Γ by assuming Γ = Δ [Waugh et al., 2004] and finding the TTD that yields the modeled partial pressure when convolved with the atmospheric history of the tracer. Note that Experiment 7 enforces saturated tracer values for CFC-11 at the surface, consistent with the assumptions of surface equilibrium used with the TTD method here, which is not the case for the control run. The difference in the inferred mean ages between the two runs thus represents a measure of the biases in mean age due to undersaturation that one might obtain when calculating Γ from observed tracer concentrations assuming 100% saturation at the surface.
 To evaluate the size of the age biases, we applied the TTD method described above to infer the average Γ of the North Pacific thermocline (σ2 = 30.15–35.35 kg m−3, i.e., all isopycnals that outcrop in the open North Pacific in the model) using the modeled CFC-11, CFC-12, and SF6 concentrations in January of model years 1980–2010 independently of each other. Each tracer shows a consistent bias toward older ages when comparing the control run to Experiment 7 (not shown). However, the percent differences in the average Γ are small (<5% for all years 1980–2010) because not all of the thermocline ventilates in areas associated with high undersaturations.
 Larger differences between the control run and Experiment 7 may be expected when inferring the ages of mode waters because mode water formation sites, including those in the North Pacific, correspond to areas of high undersaturation in late winter (see section 3.1). We estimated the differences in average Γ in the North Pacific thermocline using only model grid points where NPCMW is present (σ2 = 33.29–34.58 kg m−3 in the model with the requirement that potential vorticity ≤ 2.5 × 10−10 m−1s−1, similar to Nakamura  and Ladd and Thompson ). We infer the mean age at model year 1980 (corresponding to the early part of the WOCE era) and 2000 (just after the atmospheric downturn in CFC-11 concentration and prior to the reduction of CFC-12; Figure 1). To reiterate, both TTD calculations assume a saturated boundary condition meaning that surface undersaturations in the control run would cause TTD mean ages to be biased. The resulting bias toward older ages of the average Γ inferred from the control run compared to Experiment 7 is significantly more pronounced for NPCMW (Table 2) than the rest of the thermocline for all three tracers. For SF6, the difference between the two runs amounts to 0.8 years (10.4%) in 1980 and 1.0 years (12.7%) in 2000. For CFCs, the differences in age are somewhat larger, amounting to 1.0 years (13.8%) in 1980 and 1.8 years (23.9%) in 2000 for CFC-11 and 1.0 years (13.7%) in 1980 and 1.6 years (19.2%) in 2000 for CFC-12. The larger differences for CFCs particularly in 2000 that occur despite a decrease in mixed layer undersaturation over time (Figure 4a) may be explained by the reduction of atmospheric CFC growth rates since ∼1990 which makes the ocean interior age calculations much more sensitive to small variations in tracer concentrations and assumptions about surface saturation. Also, atmospheric CFC concentrations in the past two decades or so are still greater than in the earlier part of the record, yielding a greater absolute difference in mixed layer concentrations between Experiment 7 and the control run despite lower percent undersaturation.
 A drift in ages from 1980 to 2000 exists even when using a saturated boundary condition in the model (Experiment 7) which is consistent with the assumption of 100% saturation used when we inferred the TTDs. However, this drift is smaller than the differences between Experiment 7 and the control run for CFCs (0.2 years for CFC-11 and 0.4 years for CFC-12; Table 2), but of similar magnitude for SF6 (−0.9 years; Table 2). The drift results from the 1-D Inverse Gaussian with Δ/Γ = 1 being a good, but not perfect approximation of TTDs in the North Pacific thermocline [Waugh et al., 2003; Sonnerup et al., 2013]. The results of our first-order TTD age comparison for NPCMW nevertheless illustrate the biases toward older ages that occur when neglecting surface undersaturations and that, particularly for CFCs, increase over time. Tracer-inferred estimates of age for other mode waters and water masses with high tracer undersaturations at their formation sites are expected to be similarly affected.
 Last, following Hall et al.  and Waugh et al. , we convolved the TTDs inferred above with the anthropogenic carbon perturbation in the atmosphere to estimate anthropogenic carbon concentrations (Cant) in NPCMW. The differences in Cant inventory between Experiment 7 and the control run are small (<3%) both in 1980 and 2000 (Table 2) indicating that despite uncertainties associated with saturation levels, CFCs and SF6 are a valuable proxy for anthropogenic CO2 in the oceans. This relative insensitivity of Cant inventories to tracer saturation levels likely arises because the absolute difference in age is small resulting in similar TTDs. Cant inventories of older mode waters with a similar percent difference (accordingly, higher absolute difference) in age due to the undersaturated mixed layer boundary are likely to be more affected. Further work allowing Pe to vary by constraining TTDs using CFCs and SF6 combined [Sonnerup et al., 2008, 2013] or utilizing the maximum entropy method [Holzer et al., 2002] will help to investigate the effects of undersaturations on tracer-inferred TTDs and Cant inventories in more detail.
 We have demonstrated that the mixed layer boundary conditions of CFC-11, CFC-12, and SF6 gases used to diagnose oceanic ventilation have both spatial and temporal variability in their gas saturation states because gas exchange can fail to keep pace with equilibrium conditions. Significant levels of undersaturation of CFCs and SF6 can occur particularly as the mixed layers deepen and cool during winter. Using the North Pacific as an example, our results suggest that these undersaturations can be explained primarily by the entrainment of low tracer concentration waters from below the mixed layer and by the seasonal cycle in the temperature-dependent gas solubilities, with the latter being the dominant cause for CFCs since ∼1985. The magnitude of the gas transfer velocity also plays an important role, suggesting that model studies (such as OCMIP-2) that scale kw to a larger global mean value than recently proposed by Sweeney et al.  might be underestimating the degree of disequilibria that CFCs, SF6, and other gases experience in the mixed layer.
 The undersaturated mixed layer boundary affects tracer-inferred quantities introducing differences of around 6–13% in global inventories of CFCs and SF6 (1970–2000) compared to a saturated boundary. For NPCMW in the model, mixed layer undersaturations introduce biases as large as ∼20% in average CFC-derived mean TTD ages (Δ/Γ = 1) in 2000, but small differences in Cant inventory due to the relatively young age of the water.
 This work was supported by grants from NSF (OCE-0825095) and NOAA (NA08OAR4310529). We thank David Darr who provided computing and programming support for the Offtrac model, David Trossman for further modeling help, and Mark Warner for insightful discussions about North Pacific CFC observations.