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See detailInverse estimates of the oceanic sources and sinks of natural CO2 and the implied oceanic carbon transport
Mikaloff Fletcher, S. E.; Gruber, N.; Jacobson, A. R. et al

in Global Biogeochemical Cycles (2007), 21(1),

We use an inverse method to estimate the global-scale pattern of the air-sea flux of natural CO2, i.e., the component of the CO2 flux due to the natural carbon cycle that already existed in preindustrial ... [more ▼]

We use an inverse method to estimate the global-scale pattern of the air-sea flux of natural CO2, i.e., the component of the CO2 flux due to the natural carbon cycle that already existed in preindustrial times, on the basis of ocean interior observations of dissolved inorganic carbon (DIC) and other tracers, from which we estimate Delta C-gasex, i.e., the component of the observed DIC that is due to the gas exchange of natural CO2. We employ a suite of 10 different Ocean General Circulation Models (OGCMs) to quantify the error arising from uncertainties in the modeled transport required to link the interior ocean observations to the surface fluxes. The results from the contributing OGCMs are weighted using a model skill score based on a comparison of each model's simulated natural radiocarbon with observations. We find a pattern of air-sea flux of natural CO2 characterized by outgassing in the Southern Ocean between 44 degrees S and 59 degrees S, vigorous uptake at midlatitudes of both hemispheres, and strong outgassing in the tropics. In the Northern Hemisphere and the tropics, the inverse estimates generally agree closely with the natural CO2 flux results from forward simulations of coupled OGCM-biogeochemistry models undertaken as part of the second phase of the Ocean Carbon Model Intercomparison Project (OCMIP-2). The OCMIP-2 simulations find far less air-sea exchange than the inversion south of 20 degrees S, but more recent forward OGCM studies are in better agreement with the inverse estimates in the Southern Hemisphere. The strong source and sink pattern south of 20 degrees S was not apparent in an earlier inversion study, because the choice of region boundaries led to a partial cancellation of the sources and sinks. We show that the inversely estimated flux pattern is clearly traceable to gradients in the observed Delta C-gasex, and that it is relatively insensitive to the choice of OGCM or potential biases in Delta C-gasex. Our inverse estimates imply a southward interhemispheric transport of 0.31 +/- 0.02 Pg C yr(-1), most of which occurs in the Atlantic. This is considerably smaller than the 1 Pg C yr(-1) of Northern Hemisphere uptake that has been inferred from atmospheric CO2 observations during the 1980s and 1990s, which supports the hypothesis of a Northern Hemisphere terrestrial sink. [less ▲]

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See detailImpact of circulation on export production, dissolved organic matter, and dissolved oxygen in the ocean: Results from Phase II of the Ocean Carbon-cycle Model Intercomparison Project (OCMIP-2)
Najjar, R. G.; Jin, X.; Louanchi, F. et al

in Global Biogeochemical Cycles (2007), 21(3),

Results are presented of export production, dissolved organic matter (DOM) and dissolved oxygen simulated by 12 global ocean models participating in the second phase of the Ocean Carbon-cycle Model ... [more ▼]

Results are presented of export production, dissolved organic matter (DOM) and dissolved oxygen simulated by 12 global ocean models participating in the second phase of the Ocean Carbon-cycle Model Intercomparison Project. A common, simple biogeochemical model is utilized in different coarse-resolution ocean circulation models. The model mean (+/- 1 sigma) downward flux of organic matter across 75 m depth is 17 +/- 6 Pg C yr(-1). Model means of globally averaged particle export, the fraction of total export in dissolved form, surface semilabile dissolved organic carbon (DOC), and seasonal net outgassing (SNO) of oxygen are in good agreement with observation-based estimates, but particle export and surface DOC are too high in the tropics. There is a high sensitivity of the results to circulation, as evidenced by (1) the correlation of surface DOC and export with circulation metrics, including chlorofluorocarbon inventory and deep-ocean radiocarbon, (2) very large intermodel differences in Southern Ocean export, and (3) greater export production, fraction of export as DOM, and SNO in models with explicit mixed layer physics. However, deep-ocean oxygen, which varies widely among the models, is poorly correlated with other model indices. Cross-model means of several biogeochemical metrics show better agreement with observation-based estimates when restricted to those models that best simulate deep-ocean radiocarbon. Overall, the results emphasize the importance of physical processes in marine biogeochemical modeling and suggest that the development of circulation models can be accelerated by evaluating them with marine biogeochemical metrics. [less ▲]

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See detailImpact of a Greenland deglaciation on climate during the next millennia
Driesschaert, Emmanuelle; Brovkin, Victor; Fichefet, Thierry et al

Conference (2006, April 04)

A new Earth system model of intermediate complexity, LOVECLIM, has been developed in order to study long-term future climate changes. It includes an interactive Greenland and Antarctic ice sheet model ... [more ▼]

A new Earth system model of intermediate complexity, LOVECLIM, has been developed in order to study long-term future climate changes. It includes an interactive Greenland and Antarctic ice sheet model (AGISM) as well as an oceanic carbon cycle model (LOCH). Those climatic components can have a great impact on future climate. The few studies in recent literature assessing the impact of polar ice sheets on future climate draw very different conclusions, which shows the need for developing such a model. A set of numerical experiments have been performed in order to study the possible perturbations of climate induced by human activities over the next millennia. A particular attention is given to the Greenland ice sheet. In most of the projections, the Greenland ice sheet undergoes a continuous reduction in volume, leading to an almost total disappearance in the most pessimistic scenarios. The impact of the Greenland deglaciation on climate has therefore been assessed through a sensitivity experiment using the scenario SRES A2. The removal of the Greenland ice sheet is responsible for a regional amplification of the global warming inducing a total melt of Arctic sea ice in summer. The freshwater flux from Greenland generates large salinity anomalies in the North Atlantic Ocean that reduce the rate of North Atlantic Deep Water formation, slowing down slightly the oceanic thermohaline circulation. [less ▲]

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See detailFuture ocean carbon cycle: a study of feedbacks with the LOVECLIM model
Mouchet, Anne ULg; Driesschaert, E.; Brovkin, V. et al

Poster (2006, February)

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See detailInverse estimates of anthropogenic CO2 uptake, transport, and storage by the ocean
Mikaloff Fletcher, S. E.; Gruber, N.; Jacobson, A. R. et al

in Global Biogeochemical Cycles (2006), 20(2),

[1] Regional air-sea fluxes of anthropogenic CO2 are estimated using a Green's function inversion method that combines data-based estimates of anthropogenic CO2 in the ocean with information about ocean ... [more ▼]

[1] Regional air-sea fluxes of anthropogenic CO2 are estimated using a Green's function inversion method that combines data-based estimates of anthropogenic CO2 in the ocean with information about ocean transport and mixing from a suite of Ocean General Circulation Models (OGCMs). In order to quantify the uncertainty associated with the estimated fluxes owing to modeled transport and errors in the data, we employ 10 OGCMs and three scenarios representing biases in the data-based anthropogenic CO2 estimates. On the basis of the prescribed anthropogenic CO2 storage, we find a global uptake of 2.2 +/- 0.25 Pg C yr(-1), scaled to 1995. This error estimate represents the standard deviation of the models weighted by a CFC-based model skill score, which reduces the error range and emphasizes those models that have been shown to reproduce observed tracer concentrations most accurately. The greatest anthropogenic CO2 uptake occurs in the Southern Ocean and in the tropics. The flux estimates imply vigorous northward transport in the Southern Hemisphere, northward cross-equatorial transport, and equatorward transport at high northern latitudes. Compared with forward simulations, we find substantially more uptake in the Southern Ocean, less uptake in the Pacific Ocean, and less global uptake. The large-scale spatial pattern of the estimated flux is generally insensitive to possible biases in the data and the models employed. However, the global uptake scales approximately linearly with changes in the global anthropogenic CO2 inventory. Considerable uncertainties remain in some regions, particularly the Southern Ocean. [less ▲]

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See detailFuture anthropogenic emissions and climate change impact on the carbon cycle; a study with the LOVECLIM model
Mouchet, Anne ULg; Driesschaert, E.; Fichefet, T. et al

Conference (2005, May)

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See detailAnthropogenic ocean acidification over the twenty-first century and its impact on calcifying organisms
Orr, James C.; Fabry, Victoria J.; Aumont, Olivier et al

in Nature (2005), 437(7059), 681-686

Today's surface ocean is saturated with respect to calcium carbonate, but increasing atmospheric carbon dioxide concentrations are reducing ocean pH and carbonate ion concentrations, and thus the level of ... [more ▼]

Today's surface ocean is saturated with respect to calcium carbonate, but increasing atmospheric carbon dioxide concentrations are reducing ocean pH and carbonate ion concentrations, and thus the level of calcium carbonate saturation. Experimental evidence suggests that if these trends continue, key marine organisms - such as corals and some plankton - will have difficulty maintaining their external calcium carbonate skeletons. Here we use 13 models of the ocean - carbon cycle to assess calcium carbonate saturation under the IS92a 'business-as-usual' scenario for future emissions of anthropogenic carbon dioxide. In our projections, Southern Ocean surface waters will begin to become undersaturated with respect to aragonite, a metastable form of calcium carbonate, by the year 2050. By 2100, this undersaturation could extend throughout the entire Southern Ocean and into the subarctic Pacific Ocean. When live pteropods were exposed to our predicted level of undersaturation during a two-day shipboard experiment, their aragonite shells showed notable dissolution. Our findings indicate that conditions detrimental to high-latitude ecosystems could develop within decades, not centuries as suggested previously. [less ▲]

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See detailAragonite Undersaturation in the High-Latitude Surface Ocean Within the 21st Century
Orr, J. C.; Fabry, Véronique ULg; Aumont, O. et al

Conference (2004, December)

The surface ocean is everywhere saturated with calcium carbonate (CaCO[SUB]3[/SUB]). Yet increasing atmospheric CO[SUB]2[/SUB] reduces ocean pH and carbonate ion concentration and thus the level of ... [more ▼]

The surface ocean is everywhere saturated with calcium carbonate (CaCO[SUB]3[/SUB]). Yet increasing atmospheric CO[SUB]2[/SUB] reduces ocean pH and carbonate ion concentration and thus the level of saturation. Here we show with ocean data and models that due to this anthropogenic acidification, some surface waters will become undersaturated within decades. When atmospheric CO[SUB]2[/SUB] reaches 550 ppmv, in year 2050 under the IS92a business-as-usual scenario, Southern Ocean surface waters begin to become undersaturated with respect to aragonite, a metastable form of CaCO[SUB]3[/SUB]. By 2100 as atmospheric CO[SUB]2[/SUB] reaches 788 ppmv, undersaturation extends throughout the entire Southern Ocean (<60[SUP]o[/SUP]S) and into the surbarctic Pacific. Meanwhile, Weddell Sea surface waters also become undersaturated with respect to calcite, the stable form of CaCO[SUB]3[/SUB]. These transient changes are much larger than seasonal, interannual, and decadal variability. They threaten high-latitude aragonite secreting organisms including cold-water corals, which provide essential fish habitat, and shelled pteropods, i.e., zooplankton that serve as an abundant food source for marine predators. [less ▲]

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See detailOn the Robustness of Air-Sea Flux Estimates of Carbon Dioxide from Ocean Inversions
Mikaloff Fletcher, S. E.; Gruber, N. P.; Jacobson, A. et al

Conference (2004, December)

Inverse methods analogous to those used for atmospheric inversions have been adapted to estimate regional air-sea fluxes of carbon dioxide using ocean interior observations of dissolved inorganic carbon ... [more ▼]

Inverse methods analogous to those used for atmospheric inversions have been adapted to estimate regional air-sea fluxes of carbon dioxide using ocean interior observations of dissolved inorganic carbon and related tracers and an Ocean General Circulation Model (OGCM). We estimate seperately the preindustrial component and the component due to the anthropogenic perturbation of atmospheric carbon dioxide. Previous sensitivity studies have shown that model circulation is one of the most important sources of error in the ocean inversion. We present estimates of preindustrial and anthropogenic air-sea carbon dioxide exchange using a suite of nine different OGCM's in order to quantify the robustness of our results and explore the role of different representations of ocean circulation in the inversion. Most of the large scale features of the inverse estimates are robust across all models. The preindustrial inverse estimates generally follow the expected pattern of uptake at high latitudes and out gassing in the tropics; however, all of the models call for out gassing in the Southern Ocean between 44S and 58 S. The greatest anthropogenic carbon uptake occurs at mid- to high- latitudes, with a large anthropogenic carbon sink in the Southern Ocean, while the bulk of the anthropogenic carbon storage occurs at mid-latitudes. Preliminary results also suggest interesting, robust differences between these inverse estimates and estimates from forward model simulations. Both the preindustrial and anthropogenic carbon dioxide flux estimates are most robust at mid and high northern latitudes, except for the high latitude North Atlantic. The carbon dioxide flux estimates are most uncertain in the Southern Ocean, where the inverse estimates are strongly dependent on the rates of deep water ventilation in the OGCM. The preindustrial inverse estimates for the Indian Ocean are also sensitive to the choice of OGCM, and the anthropogenic estimates have significant uncertainties in the tropical Pacific. Over large spatial scales, inverse estimates based on different OGCM's are in better agreement than estimates based on forward simulations of the same models, but this is not necessarily true for smaller model regions. [less ▲]

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See detailEvaluation of OCMIP-2 ocean models' deep circulation with mantle helium-3
Dutay, J. C.; Jean-Baptiste, P.; Campin, J. M. et al

in Journal of Marine Systems (2004), 48(1-4), 15-36

We compare simulations of the injection of mantle helium-3 into the deep ocean from six global coarse resolution models which participated in the Ocean Carbon Model Intercomparison Project (OCMIP). We ... [more ▼]

We compare simulations of the injection of mantle helium-3 into the deep ocean from six global coarse resolution models which participated in the Ocean Carbon Model Intercomparison Project (OCMIP). We also discuss the results of a study carried out with one of the models, which examines the effect of the subgrid-scale mixing parameterization. These sensitivity tests provide useful information to interpret the differences among the OCMIP models and between model simulations and the data. We find that the OCMIP models, which parameterize subgrid-scale mixing using an eddy-induced velocity, tend to underestimate the ventilation of the deep ocean, based on diagnostics with delta(3)He. In these models, this parameterization is implemented with a constant thickness diffusivity coefficient. In future simulations, we recommend using such a parameterization with spatially and temporally varying coefficients in order to moderate its effect on stratification. The performance of the models with regard to the formation of AABW confirms the conclusion from a previous evaluation with CFC-11. Models coupled with a sea-ice model produce a substantial bottom water formation in the Southern Ocean that tends to overestimate AABW ventilation, while models that are not coupled with a sea-ice model systematically underestimate the formation of AABW We also analyze specific features of the deep He-3 distribution (He-3 plumes) that are particularly well depicted in the data and which put severe constraints on the deep circulation. We show that all the models fail to reproduce a correct propagation of these plumes in the deep ocean. The resolution of the models may be too coarse to reproduce the strong and narrow currents in the deep ocean., and the models do not incorporate the geothermal heating that may also contribute to the generation of these currents. We also use the context of OCMIP-2 to explore the potential of mantle helium-3 as a tool to compare and evaluate modeled deep-ocean circulations. Although the source function of mantle helium is known with a rather large uncertainty, we find that the parameterization used for the injection of mantle helium-3 is sufficient to generate realistic results, even in the Atlantic Ocean where a previous pioneering study [J. Geophys. Res. 100 (1995) 3829] claimed this parameterization generates inadequate results. These results are supported by a multi-tracer evaluation performed by considering the simulated distributions of both helium-3 and natural C-14, and comparing the simulated tracer fields with available data. (C) 2004 Elsevier B.V. All rights reserved. [less ▲]

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See detailEvaluation of ocean carbon cycle models with data-based metrics
Matsumoto, K.; Sarmiento, J. L.; Key, R. M. et al

in Geophysical Research Letters (2004), 31(7),

New radiocarbon and chlorofluorocarbon-11 data from the World Ocean Circulation Experiment are used to assess a suite of 19 ocean carbon cycle models. We use the distributions and inventories of these ... [more ▼]

New radiocarbon and chlorofluorocarbon-11 data from the World Ocean Circulation Experiment are used to assess a suite of 19 ocean carbon cycle models. We use the distributions and inventories of these tracers as quantitative metrics of model skill and find that only about a quarter of the suite is consistent with the new data-based metrics. This should serve as a warning bell to the larger community that not all is well with current generation of ocean carbon cycle models. At the same time, this highlights the danger in simply using the available models to represent the state-of-the-art modeling without considering the credibility of each model. [less ▲]

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See detailEvaluating global ocean carbon models: The importance of realistic physics
Doney, S. C.; Lindsay, K.; Caldeira, K. et al

in Global Biogeochemical Cycles (2004), 18(3),

A suite of standard ocean hydrographic and circulation metrics are applied to the equilibrium physical solutions from 13 global carbon models participating in phase 2 of the Ocean Carbon-cycle Model ... [more ▼]

A suite of standard ocean hydrographic and circulation metrics are applied to the equilibrium physical solutions from 13 global carbon models participating in phase 2 of the Ocean Carbon-cycle Model Intercomparison Project (OCMIP-2). Model-data comparisons are presented for sea surface temperature and salinity, seasonal mixed layer depth, meridional heat and freshwater transport, 3-D hydrographic fields, and meridional overturning. Considerable variation exists among the OCMIP-2 simulations, with some of the solutions falling noticeably outside available observational constraints. For some cases, model-model and model-data differences can be related to variations in surface forcing, subgrid-scale parameterizations, and model architecture. These errors in the physical metrics point to significant problems in the underlying model representations of ocean transport and dynamics, problems that directly affect the OCMIP predicted ocean tracer and carbon cycle variables (e.g., air-sea CO2 flux, chlorofluorocarbon and anthropogenic CO2 uptake, and export production). A substantial fraction of the large model-model ranges in OCMIP-2 biogeochemical fields (+/-25-40%) represents the propagation of known errors in model physics. Therefore the model-model spread likely overstates the uncertainty in our current understanding of the ocean carbon system, particularly for transport-dominated fields such as the historical uptake of anthropogenic CO2. A full error assessment, however, would need to account for additional sources of uncertainty such as more complex biological-chemical-physical interactions, biases arising from poorly resolved or neglected physical processes, and climate change. [less ▲]

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See detailInverse Estimates of Anthropogenic Carbon Dioxide From Ocean Interior Carbon Measurements and Ocean General Circulation Models
Mikaloff Fletcher, S. E.; Gruber, N. P.; Jacobson, A. R. et al

Conference (2003, December)

The ocean is an important sink for atmospheric carbon dioxide, and the exchange of carbon dioxide between the atmosphere and ocean plays a critical role in determining the spatial distribution of ... [more ▼]

The ocean is an important sink for atmospheric carbon dioxide, and the exchange of carbon dioxide between the atmosphere and ocean plays a critical role in determining the spatial distribution of atmospheric carbon dioxide. However, there is still a great deal of uncertainty in both magnitude and regional patterns of anthropogenic uptake associated with estimates of oceanic carbon fluxes. Using a recently developed technique, exchange of anthropogenic carbon dioxide across the air-sea interface have been estimated from observations of dissolved inorganic carbon and nutrient concentrations and an Ocean General Circulation Model (OGCM) using a Green's function inverse modeling technique. Previous sensitivity studies have shown that model circulation error is an important source of error in the ocean inversion. In order to address the role of ocean circulation biases, inverse estimates of anthropogenic carbon air-sea gas exchange are presented using basis functions from a suite of seven different OGCM's. The robustness of the ocean inversion will be quantified and the effects of differences between approaches to modeling ocean circulation on the ocean carbon cycle will be explored. These results will be discussed in the context of recent atmospheric inverse estimates. [less ▲]

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See detailLOVECLIM, a three-dimensional model of the Earth system for investigating long-term climate changes
Driesschaert, E.; Brovkin, V.; Fichefet, T. et al

Conference (2003, September)

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See detailLOVECLIM, a three-dimensional model of the Earth system for investigating long-term climate changes
Driesschaert, E.; Fichefet, T.; Goosse, G. et al

Poster (2003, April 08)

A three-dimensional global model of the Earth system suitable for studying the long-term evolution of climate (LOVECLIM) has been recently developed. This model is made up of a coarse-resolution three ... [more ▼]

A three-dimensional global model of the Earth system suitable for studying the long-term evolution of climate (LOVECLIM) has been recently developed. This model is made up of a coarse-resolution three-dimensional atmosphere-sea-ice-ocean model (ECBILT-CLIO), a dynamical model of the continental biosphere (VECODE), a comprehensive model of the oceanic carbon cycle (LOCH), and a high-resolution thermomechanical model of the Greenland and Antarctic ice sheets (AGISM). The atmospheric component has the big advantage that it has been simplified to a level that makes runs on a multi-century time-scale computationaly feasible, while still being capable of producing results that, on the whole, are comparable to those of atmospheric general circulation models. The performance of the coupled model is evaluated by performing ensemble simulations over the period 1500-2000 and by comparing the model results to available climate reconstructions. In these simulations, the following forcings are taken into consideration : the variations in solar irradiance, the volcanic activity, the anthropogenic emissions of CO2, and the changes in concentration of other greenhouse gases and sulphate aerosols resulting from human activities. In the future, the model will be used to investigate the evolution of climate and sea level over the third millennium. [less ▲]

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See detailA note on the age of radioactive tracers
Delhez, Eric ULg; Deleersnijder, Eric; Mouchet, Anne ULg et al

in Journal of Marine Systems (2003), 38(3-4), 277-286

wThe age of a water mass is often estimated experimentally using the radio-age computed from the distribution of a radioactive tracer (radiocarbon, helium-tritium). Deleersnijder et al. [J. Mar. Syst. 28 ... [more ▼]

wThe age of a water mass is often estimated experimentally using the radio-age computed from the distribution of a radioactive tracer (radiocarbon, helium-tritium). Deleersnijder et al. [J. Mar. Syst. 28 (2001) 229.] have shown that the radio-age underestimates the age of the water and is larger than the age of the radioactive tracer used for its evaluation. This result is generalized here to radio-ages computed from the ratio of two radioactive tracers. The differences between the different ages are also studied analytically and numerically as functions of the decay rate of the radioactive tracers. For small decay rates, the difference between the age of the water mass and the radio-age is shown to be proportional to the decay rate. It depends also on the level of mixing in the system; even radioactive tracers with small decay rates can provide poor estimates of the age of the water mass in a strongly diffusive flow. For small half lives, both the radio-age and the age of radioactive tracers decrease as the inverse of the square root of the decay rate. The same analysis applies to some extent to the estimates of the age of a water mass from stable tracers with known time dependent sources (e.g. chloroflurocarbons). (C) 2002 Elsevier Science B.V. All rights reserved. [less ▲]

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See detailCAT, The Constituent-oriented Age Theory, and its application to marine flows
Deleersnijder, Eric; Delhez, Eric ULg; Mouchet, Anne ULg et al

Conference (2002, May)

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See detailEvaluation of deep water circulation with natural C-14 and helium-3 during OCMIP-2
Dutay, J.-C; Jean-Baptiste, P; Maier-Reimer, E. et al

Conference (2002, May)

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See detailEvaluation of ocean model ventilation with CFC-11: comparison of 13 global ocean models
Dutay, J.-C.; Bullister, J. L.; Doney, S. C. et al

in Ocean Modelling (2002), 4(2), 89-120

We compared the 13 models participating in the Ocean Carbon Model Intercomparison Project (OCMIP) with regards to their skill in matching observed distributions of CFC-11. This analysis characterizes the ... [more ▼]

We compared the 13 models participating in the Ocean Carbon Model Intercomparison Project (OCMIP) with regards to their skill in matching observed distributions of CFC-11. This analysis characterizes the abilities of these models to ventilate the ocean on timescales relevant for anthropogenic CO2 uptake. We found a large range in the modeled global inventory (+/- 30\%), mainly due to differences in ventilation from the high latitudes. In the Southern Ocean, models differ particularly in the longitudinal distribution of the CFC uptake in the intermediate water, whereas the latitudinal distribution is mainly controlled by the subgrid-scale parameterization. Models with isopycnal diffusion and eddy-induced velocity parameterization produce more realistic intermediate water ventilation. Deep and bottom water ventilation also varies substantially between the models. Models coupled to a sea-ice model systematically provide more realistic AABW formation source region; however these same models also largely overestimate AABW ventilation if no specific parameterization of brine rejection during sea-ice formation is included. In the North Pacific Ocean, all models exhibit a systematic large underestimation of the CFC uptake in the thermocline of the subtropical gyre, while no systematic difference toward the observations is found in the subpolar gyre. In the North Atlantic Ocean, the CFC uptake is globally underestimated in subsurface. In the deep ocean, all but the adjoint model, failed to produce the two recently ventilated branches observed in the North Atlantic Deep Water (NADW). Furthermore, simulated transport in the Deep Western Boundary Current (DWBC) is too sluggish in all but the isopycnal model, where it is too rapid. (C) 2002 Elsevier Science Ltd. All rights reserved. [less ▲]

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