We employ three different models, two of which use an offline approach and one is an online fully coupled earth system model. For the offline models we use the transport matrix method (TMM) described in detail by Khatiwala et al. (2005) and Khatiwala (2007). In this approach ocean-tracer transport is represented by a matrix operation involving the tracer field and a transport matrix extracted from a global circulation online model (Khatiwala, 2007). In particular, we use two matrices extracted from two versions of the MIT (Massachusetts Institute of Technology) general-circulation model, a state-of-the-art primitive-equation model (Marshall et al., 1997). The coarse-resolution matrix (hereafter MIT2.8) was derived from a 2.8∘×2.8∘ global configuration of this model with 15 vertical layers, forced with monthly mean climatological fluxes of momentum, heat and freshwater, and subject to a weak restoring of surface temperature and salinity to observations. The higher resolution matrix (hereafter ECCO) is based on the data-assimilation model of the ECCO consortium (Estimating the Circulation and Climate of the Ocean; Stammer et al., 2004) and has a horizontal resolution of 1∘×1∘ and 23 vertical layers; for details see Khatiwala (2007) and Kriest et al. (2010, 2012). Wind-speed dependence of gas exchange applies winds from Trenberth et al. (1989) with a monthly resolution regridded to the respective model grid. Sea ice fields applied are the OCMIP-2 ice mask (Orr et al., 2000) for MIT2.8 and NASA ISLSCP (International Satellite Land Surface Climatology Project) climatology (http://iridl.ldeo.columbia.edu/SOURCES/.NASA/.ISLSCP/.GDSLAM/.Snow-Ice-Oceans/.sea/.sea_ice/) for ECCO (S. Dutkiewics, MIT, personal communication, 2011). OCMIP is the Ocean Carbon cycle Model Intercomparison Project (http://ocmip5.ipsl.jussieu.fr/OCMIP/).

The third model used is the University of Victoria Earth System Climate Model (UVIC; Weaver et al., 2001), version 2.8 in the configuration used at the GEOMAR Helmholtz Centre for Ocean Research, Kiel, Germany (Oschlies et al., 2008). The ocean component of this model is a coarse-resolution (1.8∘×3.6∘, 19 vertical layers) 3-D ocean general-circulation model (Modular Ocean Model, Version2). Wind velocities are prescribed from the NCAR/NCEP (National Center for Atmospheric Research of the United States National Centers for Environmental Prediction of) monthly climatology. Sea ice coverage is computed from a dynamic/thermodynamic sea ice model (Bitz et al., 2001). The biogeochemical ocean model of the UVIC is described in detail by Schmittner et al. (2008).

For the 14C simulations with the TMM models, we largely follow the OCMIP-2 protocol (Orr et al., 2000; Jahn et al., 2014) and study the natural 14C distribution in an abiotic setting and against an atmosphere of Δ14C=0 and a constant pCO2atm=280 µatm. DIC and 14C-DIC are prognostic model tracers of total dissolved inorganic carbon and its 14C isotope, respectively. Alkalinity is prescribed from the model's salinity field assuming a fixed alkalinity / salinity ratio. OCMIP-2 14C simulations are abiotic model runs; biotic fluxes of 14C (as well as of DIC and alkalinity) are ignored following Fiadeiro (1982). Also the effect of isotope fractionation is not considered. Our notation of Δ14C follows the OCMIP-2 protocol (Orr et al., 2000).

All model runs were integrated for several thousand years (for details see Sect. 3) and can be considered equilibrium runs. For UVIC the 14C simulations can be made alongside a  biotic model run (Schmittner et al., 2008).

Air–sea exchange of CO2 and 14CO2 in all three models is treated according to Eqs. (1) and (2): CO2(ex)=(1-ice)⋅kw⋅(CO2∗(water)-CO2∗(air)),14CO2(ex)=(1-ice)⋅kw⋅(CO2∗(water)⋅R(water)-CO2∗(air)⋅R(atm)), kw=a⋅Un⋅(Sc/660)-η, where CO2∗(air)=CO2(sol)⋅pCO2(atm)⋅P(atm).

kw is the gas-transfer velocity, U is wind speed, n=2, Sc is the Schmidt number, and η=0.5. CO2∗(water) is the sum of CO2 dissolved in seawater and H2CO3 in surface water computed from the DIC concentration and an estimate of pH (e.g. Follows et al., 2006). CO2∗(air) is the equilibrium CO2 concentration given atmospheric pCO2, CO2 solubility and the local atmospheric pressure; CO2(sol) is the solubility of CO2, pCO2(atm) is CO2 partial pressure in the atmosphere, P(atm) is the local atmospheric pressure, R(atm) is the 14C / C ratio of the atmosphere and R(water) is the 14C / C ratio of the surface water. In the standard configuration the gas-transfer velocity kw is computed using a value of a=0.337, following the OCMIP-2 protocol. The term “ice” represents the fraction of water area covered by sea ice.

In the ocean, bulk 14C-age (in units of years) can be computed (Stuiver and Polach, 1977) according to Eq. (3): 14C-age=-8267log⁡e(Δ14C/1000+1).

In order to study the distribution of preformed 14C in the interior of the ocean, we designed a suite of additional model tracers.

14C-DICbulk: this is the tracer of natural 14C-DIC implemented following the OCMIP-2 protocol. The age computed from this tracer via Eq. (3) has also been called “conventional 14C-age” (Khatiwala et al., 2012) but is usually referred to as “14C-age” or “radiocarbon age”. We will use the term 14C-agebulk in order to highlight the fact that it consists of several components (see below).

Reference model runs (10 000 yr) are carried out with all three models. We apply a gas-transfer constant of a=0.337, wind fields and ice cover as given in Sect. 2.1 for these runs. Implemented tracers are DIC, 14C-DICbulk and ageideal. We use these tracers to approximate the preformed component of 14C-agebulk in the different models by diagnosing it during post-processing from the difference of 14C-agebulk and ageideal. Reference runs also serve as spin-up runs from which other model experiments are initialised.

To start with, we compare global mean profiles of 14C-agebulk and ageideal (Fig. 2a). A number of features are evident. First, 14C-agebulk is much larger than ageideal in any model. The global mean offset between the two age measures varies by up to a factor of 2 between models (Fig. 2b). In the deep ocean the offset is about 400 yr in MIT2.8 and 680 yr (800 yr) in ECCO (UVIC). The age offset may be either rather homogeneous vertically (MIT2.8) or have a marked vertical gradient of up to 400 yr difference between surface and deep water (ECCO and UVIC). Second, global mean surface 14C-agebulk is smaller than the data-based estimate from the Global Ocean Data Analysis Project (GLODAP) in all three models (Fig. 2a). Third, a judgement based just on global mean profiles of 14C-agebulk would indicate that over most of the ocean the UVIC model is the one in best agreement with observations. Furthermore, one might conclude that the MIT2.8 model appears to have too young waters and presumably too vigorous a circulation almost everywhere.

Interestingly, the ageideal tracer indicates just the opposite. Deep-ocean MIT2.8 waters have the highest ages pointing to a more sluggish circulation while in UVIC (and ECCO) deep-ocean waters are in fact younger, indicating a more vigorous circulation compared to the MIT2.8 model. Finally, in the upper 2000 m, the global mean profiles of the ideal age tracer suggest that the circulations are similar in all three models, at least much more similar than indicated by 14C-agebulk.

In conjunction with the observations that ocean–atmosphere 14C equilibration is slow (Broecker and Peng, 1974) and surface-ocean 14C-agebulk is well above zero (see Fig. 1b), we suspect that (most of) the difference between 14C-agebulk and ageideal in the interior of the ocean is due to the 14C-agebulk which a water mass had at the time when entering the ocean's interior, i.e. its preformed 14C-age.

In Fig. 3, we present the large-scale distribution of 14C-agebulk and ageideal from the ECCO model run along sections through the Atlantic Ocean (20∘ W) and the Pacific Ocean (140∘ W). Surface 14C-agebulk is around 200 yr in the subtropical ocean basins, around 300 yr in the North Atlantic and around 1000 yr in the Southern Ocean. In the interior of the ocean, 14C-agebulk increases from about 300 yr in the northern North Atlantic Ocean, almost continuously along the path of circulation originally proposed for the global conveyor belt by Broecker and Peng (1982), towards the deep northern North Pacific where 14C-agebulk is about 2000 yr (Fig. 3a). In contrast, ageideal (Fig. 3b) is zero all over the surface ocean, and close to zero in the deep waters of the two major ocean ventilation regions, i.e. the northern North Atlantic and the Southern Ocean (Marshall and Speer, 2012). Elevated ageideal is found in the deepest waters of the Atlantic Ocean (700 yr) and in particular towards the northern North Pacific where maximum ages are around 1400 yr along the transect chosen. Basin-scale patterns of ageideal and 14C-agebulk are similar in the Pacific mainly due to a very homogeneous N–S distribution of preformed age (Fig. 3c). In the Atlantic Ocean, however, the strong N–S gradient in preformed age masks important aspects of circulation in the 14C-agebulk distribution. For example, the continuous north-to-south increase in the 14C-agebulk is not consistent with the strong ventilation in the Southern Ocean, but is mainly governed by waters of large preformed 14C-age subducting in the Atlantic Sector of the Southern Ocean.

The preformed age shown in Fig. 3c is taken from the agepre tracer (Table 1). The sum of ageideal and this preformed age tracer agrees with the 14C-agebulk within a few percent (see Fig. 3d for the residual). As we will explain and quantify in the following section, the residual derives from a non-linear effect of 14C-DIC and DIC tracer mixing on computed age. To reflect this we write Eq. (4):

14C-agebulk=ageideal+14C-agepre+“mixing residual”.

In the following we will use dedicated model experiments carried out with the ECCO model, in order to quantify the relative importance of the three terms on the right-hand side of Eq. (4). We implement DIC and all six tracers described in Sect. 2.2. We will use this combination of tracers to quantify the non-linearity arising from mixing of the 14C-DIC and DIC tracers on computed 14C-age components. To explore the effect of tracer mixing on 14C-ages in more detail, we first apply the additional tracers 14C-DICpre and 14C-DICdecay (Table 1). We initialise these tracers from the model output of the spin-up run (after 4000 yrs) with the DIC, 14C-DIC and ageideal tracers assuming the “mixing residual” term of Eq. (4) to be zero everywhere. Running the model for another 6000 yr, we find the sum of the preformed and the decay tracers to match the 14C-DIC tracer perfectly (Fig. 4a). The sum of ages (14C-agepre+14C-agedecay), however, is smaller by 6 % on average than the age computed from the 14C-DICbulk tracer (Fig. 4b).

The difference between Fig. 4a and b, i.e. the low bias in ages computed from 14C tracers relative to the tracer itself, is explained by the combination of the logarithmic transformation in the age computation (Eq. 3) and the effect of mixing of waters with different 14C / C tracer ratios (Jenkins, 1987; Delhez et al., 2003; Khatiwala et al., 2001, 2012).

To make this effect visible and quantifiable in our model, we compare age estimates from two sets of tracers (Table 1) tracking (a) the circulation component of age, (b) preformed age, and (c) bulk age. One set of these tracers behaves ideally in the interior of the ocean, in the sense that where they are affected by mixing, the mixing products can be described by mixing along a linear mixing line. These tracers are the ageideal, agepre and agebulk tracers (Table 1). The latter two tracers inherit the age of 14C-agebulk at the surface, while in the interior of the ocean they behave like ideal tracers, being either only transported (agepre tracer) or being both transported and ageing with a rate of 1 dayday-1 (agebulk tracer). We compare ages derived from these ideally behaving tracers and the respective ages from the 14C-based tracers, 14C-agedecay, 14C-agepre and 14C-agebulk. In all three cases (circulation component of age, preformed component of age and bulk age) we see that 14C-based ages underestimate their ideally behaving counterparts. We present the results as anomalies (ideally behaving – 14C-based) of ages along the combined section through the Atlantic (30∘ W), Southern (60∘ S) and Pacific (140∘ W) oceans (Fig. 5). The age anomaly agebulk-14C-agebulk (Fig. 5a) is close to zero in the surface ocean, in the northern North Atlantic, and in the Atlantic sector of the Southern Ocean. Away from these outcrop regions and largely following increasing ideal age (ageideal), the anomaly increases to maximum values of about 50 yr in the (South) Atlantic Ocean and about 80 yr in the (North) Pacific Ocean. This difference is moderate and equivalent to a few percent of 14C-agebulk. Preformed ages (Fig. 5b) show very small anomalies (agepre-14C-agepre) of only a few years (and usually less than 1 % of agepre), again with maxima in the South Atlantic Ocean and the North Pacific Ocean. The largest difference is found between ageideal and 14C-agedecay. In the deep northern North Pacific this difference is almost 200 yr (Fig. 5c). Over much of the Pacific Ocean it is equivalent to about 15 % of ideal age.

The effect of non-linear mixing on 14C-ages has been studied previously (Deleersnijder et al., 2001; Holzer et al., 2010; Khatiwala et al., 2012). Applying a boundary propagator approach (Holzer et al., 2010), Khatiwala et al. (2012) found a difference between their mean age (Γ) and their radiocarbon age (ΓC) of usually less than 50 yr, which is comparable to the overall effect of non-linear mixing (agebulk - 14C-agebulk) (Fig. 6a) observed in our model, while the difference (ageideal - 14C-agedecay) from our model (Fig. 6c) is considerably larger. This may be explained by methodological differences. While our 14C-agedecay is based on the numerical integration of 14C decay, the definition of the radiocarbon age, 14C(x) = 14C0(x)eλΓC(x), of Khatiwala et al. (2012) uses 14C0 as the weighted average of the 14C surface concentration.

The small difference agebulk-14C-agebulk (Fig. 5a) in combination with an almost perfect behaviour of the preformed-age tracers (Fig. 5b) suggests that our initial assumption (Sect. 3.1; Fig. 2) that the preformed age can be well approximated by Eq. (5), i.e. the difference between the 14C-agebulk and ageideal of a model, is justified. 14C-agepre≈14C-agebulk-ageideal

In any case, preformed 14C-ages estimated from this difference provide a conservative, lower-limit estimate of preformed age. In the ECCO model, this underestimate may be as large as 20 % in individual grid boxes (Fig. 4c). On average, however, it is about 7 % with higher values observed towards the North Pacific Ocean. This uncertainty is small given the order of 50 % contribution of the preformed age to bulk 14C-ages presented in Sect. 3.1. For the sake of saving computational time by having a reduced number of tracers, we hence ignore the mixing effect in the following section where we discuss a series of sensitivity runs.

In this section we treat the major processes, which determine the magnitude of preformed 14C-age, and how they influence model assessment if based on 14C-agebulk. We perform several sensitivity experiments to study the sensitivity of preformed 14C-age distribution to relevant model parameters. All sensitivity experiments are carried out with the reduced set of model tracers (i.e. 14C-DICbulk and ideal age tracer, Table 1) and we diagnose the preformed 14C-age offline during post-processing of model output using Eq. (5). This procedure is justified by the results presented in Sect. 3.2.

14C-agebulk of several hundred years in the surface ocean (Fig. 1b) have been attributed to the long equilibration times of carbon isotopes (Broecker and Peng, 1974). While for CO2 the equilibration time is governed by the product of the timescale of gas exchange (of the order of 1 month) and the ratio CO32-/CO2aq (10–15 in the surface ocean), the equilibration time of carbon isotopes scales with the ratio TCO2/CO2aq. Since there is about 10 times more total CO2 than there are carbonate ions in seawater, the equilibration time of carbon isotopes is larger by about a factor of 10, i.e. of the order of a decade (Broecker and Peng, 1974). Elevated and variable 14C-DICbulk in the surface ocean suggests that the residence time of waters at the ocean surface is usually much shorter than this equilibration time and equilibrium with atmospheric 14C is therefore not attained. The actual residence time (Bolin and Rohde, 1973; Takeoka, 1984) of waters in the surface ocean is not well known though. Diagnosing the residence time of surface waters, and particularly its regional variations with respect to the observed distribution of 14C-DICbulk at the surface, is not straightforward in our model. Instead, we take a first step into this direction and model the age of the surface water relative to its last stay below a given depth. For this purpose we modify the definition of our ideal age tracer such that it is set to zero everywhere below a model specific reference depth and allowed to age in layers higher up. The reference depths are 135 m in ECCO and 120 m in MIT2.8. The idea here is to have a reference depth larger than 100 m, a depth often used pragmatically to define the productive surface layer. Differences between the reference depths are simply due to the different vertical resolutions in the models. The time passed since the last residence below the surface is henceforth referred to as the “age relative to depth”. In our MIT2.8 model, for example, the age relative to depth ranges up to 2 years in subpolar and most Northern Hemisphere polar waters, up to 5 years in Southern Ocean polar waters and equatorial upwelling regions, and up to 7 years in the subtropical gyres (Fig. 6). In the ECCO model the age relative to depth in the Southern Ocean is lower. In general, in areas of deep convection or upwelling the age relative to depth is low while in areas characterized by horizontal advection and downwelling it is larger. Deep convection and upwelling are thus a continuous source of old waters low in 14C to the surface ocean. Our estimates of the age relative to depth (Fig. 6) are qualitatively consistent with the observed distribution of 14C-agebulk at the surface (Fig. 1b). Regions with low age relative to depth show high 14C-agebulk, and vice versa. Still, our age relative to depth may be considered lower estimates of true residence or exposure time (Delhez et al., 2004) since the respective age tracer will be reset to zero each time a water parcel is below the reference depth, even if for a brief period only.

Most of the deep-ocean volume is ventilated from relatively small regions in the high latitudes. It is conditions in these regions that control the preformed 14C-age distribution in the ocean's interior. One such region is the northern North Atlantic. Surface waters there, originating mainly from the low-latitude Atlantic Ocean, are to be converted into North Atlantic Deep Water (NADW). Source waters have been at or near the surface for several years allowing 14C-DIC to approach equilibrium with the atmosphere. Furthermore, deep convection in the northern North Atlantic entrains relatively young waters into the surface each winter. Combined, both effects give rise to moderately negative surface Δ14C and moderate 14C-ages in the surface (Fig. 1b).

In the Southern Ocean the situation is different. Upwelling south of the Antarctic Polar Front brings very old waters to the surface. In fact, some of this water stems from the return flow of the global conveyor belt. Having left the ocean's surface in the northern North Atlantic it has travelled through the deep Atlantic Ocean, the Circumpolar Current system, further up to the North Pacific and back to the Southern Ocean isolated from the atmosphere all the time, which has been estimated to be of the order of 2700 yr (DeVries and Primeau, 2011). Other components of the water upwelling in the Southern Ocean have been ventilated relatively recently in the North Atlantic or have returned after a passage of about 2000 yr from the tropical Indian Ocean. Hence, waters upwelling in the Southern Ocean are in bulk much older and more depleted in 14C-DIC, compared to those entering the deep-water formation regions at the surface of the North Atlantic. Combined with short surface residence times, this gives rise to much larger preformed 14C-ages in the Southern Ocean deep-water formation regions, about 1000 yr in the real ocean (Bard, 1988; Fig. 1b).

Several factors could potentially influence the overall magnitude and distribution of preformed 14C-age in models and the real ocean. These are (a) the intensity of upwelling in the Southern Ocean, (b) the rate of gas exchange, (c) ice coverage, (d) water residence time in the surface of the region of water mass formation and (e) the relative contribution of different source water regions (e.g. NADW and AABW, Antarctic Bottom Water) to the total deep-water formation rate.

The gas-exchange formulation (Eqs. 1–3) is essentially identical in all three tested models. In particular the standard configurations of all models apply wind-speed squared and the OCMIP-2 gas-transfer constant of 0.337. This value is based on tuning one model of the OCMIP-2 family together with its given wind and sea ice fields against the bomb 14C ocean inventory estimated from observations (Broecker et al., 1985) and considered correct at the time of the OCMIP-2 experiment. Evidence has since accumulated suggesting the bomb 14C ocean inventory to be in fact smaller by up to 25 % (Sweeney et al., 2007). As a consequence, the gas-transfer constant may need a corresponding reduction. Such a change in the gas-transfer constant has little effect on net oxygen or total-CO2 fluxes between ocean and atmosphere. It has, however, a considerable effect on 14C-agepre and hence also the 14C-agebulk distribution in the ocean. Using all models, we repeat the standard experiment with a reduced gas-exchange rate. For this purpose we reduce the standard value of the gas-transfer constant from a=0.337 to a value of a=0.24 (see Eq. 2). This change causes the global mean profiles of preformed 14C-ages (Fig. 7a) to increase by about 150 yr (ECCO, UVIC) to 200 yr (MIT2.8). In the global mean profile this shift is almost uniform with depth. Concerning the global mean profiles of 14C-agebulk, two features are evident (Fig. 7b). First, model surface values are now (a=0.24) much closer to bulk ages derived from the “observed” natural 14C as compared to our reference runs (a=0.337; Fig. 2a). Reducing the gas-transfer constant hence solves one of the model-data comparison issues discussed in Sect. 3.1 (Fig. 2a). At depth (ignoring the deepest layers below 4000 m) this increase shifts the global mean profile of the MIT2.8 much closer to observations. With the reduced gas-exchange constant 14C-agebulk of the UVIC model appears to be too large compared to observations and the ECCO model appears to be the best-performing model in our model inter-comparison, except at the surface. 14C-based judgement of model circulation obviously is very sensitive to the air–sea exchange formulation, which, however, only affects preformed age, not ageideal. Using an improper gas-exchange formulation may hence adversely affect the interpretation of 14C model experiments concerning a model's circulation dynamics.

One potential solution to this problem is to diagnose the most suitable gas-exchange constant for a given model and wind field by performing a bomb 14C calibration experiment (Sweeney et al., 2007). The degree to which this is possible, however, is limited by several methodological problems. The number of 14C ocean data available from early after the atomic bomb testing in the atmosphere, i.e. the 1970s (GEOSECS program, Broecker et al., 1985; see also Schlitzer, 2015) is small compared with the number of respective data from the 1990s, i.e. from the WOCE (World Ocean Circulation Experiment) and CLIVAR (Climate and Ocean: Variability, Predictability and Change) observational programs (Key et al., 2004). The bomb 14C ocean inventory of the 1970s is hence less certain than that of the 1990s. This second time slice, however, may be too late to constrain the adequate gas-exchange coefficient of a model independent of the model's ocean circulation (e.g. Graven et al., 2012) as 14C back fluxes from the ocean to the atmosphere (Naegler, 2009) become increasingly important. The separation of bomb 14C and natural 14C (Rubin and Key, 2002; Sweeney et al., 2007) as well as details of the model implementation of 14C (Mouchet, 2013) add to inevitable uncertainties of a bomb 14C calibration of the gas exchange in a given model.

Ice coverage is another factor potentially influencing the gas equilibration at deep-water formation sites (Ito et al., 2004; Duteil et al., 2013). Ito et al. (2004) reported ice cover to be responsible for about one-third of the oxygen disequilibrium observed in their model. In order to study the impact of ice cover on 14C-gas exchange and hence preformed 14C-age, we perform one model run with the ECCO model where ice cover was switched off for 6000 yr. Technically this run was initialised with data from year 4000 of the spin-up and the value of “ice” in Eq. (1) was prescribed to zero. In this experiment preformed 14C-age was reduced by up to 70 yr, or less than 10 % of its normal value (Fig. 8). Ice cover hence appears not to be of major importance in controlling preformed 14C-ages.

Campin et al. (1999) observed differences in the response of 14C-agebulk and ageideal to atmospheric forcing representing the Last Glacial Maximum (LGM) and the present-day ocean, respectively. The associated difference of 14C-agepre between LGM and today's ocean has been discussed to be related to an intensified upwelling of 14C-depleted Circumpolar Deep Water (Campin et al., 1999) or an ice-cover-induced reduction in 14C-gas exchange during the LGM (Schmittner, 2003). Since in both studies, ice cover and circulation changed simultaneously, a direct comparison with our experiments is difficult.