A fundamental measure of global warming is the heat uptake by the ocean, which represents more than 90% of the excess energy gained by the Earth1. This ocean warming has been quantified using hydrographic temperature measurements, including data from the Argo float program, which expanded coverage after 20072,3,4. As shown in Fig. 1, the most recent temperature-based estimates of ocean warming5,6,7,8 show good agreement for 2007-2016 (1.09 ± 0.10 to 1.16 ± 0.2 × 1022 J yr−1), but a larger spread when extending back to include the sparser 1990s data (0.90 ± 0.09 to 1.36 ± 0.10 × 1022 J yr−1 for 1993–2015). The spread is mostly caused by gap-filling methods and systematic errors5,8,9, which together introduce uncertainties up to 25–50% in warming trends10. Because temperature-based estimates use the same upper-ocean observations and linear warming trend for depths below 2000 m (ref. 6.), they may share additional unknown systematic errors8. An alternative method based on the top of the atmosphere energy balance11 is also not truly independent, because it is subject to large systematic errors when estimating long-term trends and therefore depends on the same hydrographic measurements for calibration11,12,13,14. Here we introduce a third method, based on changes in the abundances of gases in the atmosphere, which respond to whole-ocean warming through the temperature dependence of gas solubility in seawater. This method is not limited by data sparseness, because fast mixing in the atmosphere efficiently integrates the global ocean signal.

Figure 1
figure 1

Change in global ocean heat content (ΔOHC). (a) ΔOHC derived from hydrographic and atmospheric observations (normalized to zero in 2007, ±1-σ uncertainty). (b) Linear least-squares trends for 1991–2016, 1993–2016 and 2007–2016 (±1-σ uncertainty). Hydrography-based ΔOHC estimates combine warming rates at ocean depths of 0 to 2,000 m warming rates (Cheng and co-authors (CHEN)8, Pacific Marine Environmental Laboratory (PMEL)6, Meteorological Research Institute (MRI)5 and National Center for Environmental Information (NCEI)47) with revised deep ocean (depth of more than 2000 m) of ref. 7 (Tables S1, S2). The atmospheric-based estimate which uses observed atmospheric potential oxygen trends (ΔAPOClimate) and model-based ΔAPOClimate-to-ΔOHC ratio, does not resolve interannual variations.

Changes in ocean heat content on seasonal15 and glacial-interglacial16 time-scales have been reconstructed using measurements of noble gases in modern or ancient air. Our method is similar, but instead of relying on noble gases (e.g. Ar/N2), which lack sufficient accuracy as yet15, we rely on measurements of atmospheric O2 and CO2, which can be summed to yield a tracer “atmospheric potential oxygen” (APO) that responds to warming similar to a noble gas17. When the ocean warms, the solubility of O2 and CO2 drops, and the amount of gas lost by the ocean can be quantified with the complementary change observed in the atmosphere. Precise atmospheric O2 measurements began in 1991 (CO2 in 1958), enabling APO-based ocean heat content reconstructions that span nearly three decades18.


Atmospheric potential oxygen trend and components

APO (=O2 + 1.05 × CO2) is computed using observed atmospheric O2/N2 mole ratios and CO2 mole fractions (see Methods)18,19,20. By design, APO is insensitive to exchanges with land ecosystems, which produce changes in O2 and CO2 that largely cancel in APO owing to their approximate 1.05 O2:C oxidative ratio. Time-series measurements at remote sites show a global long-term decline in APO ΔAPOOBS = 255.93 ± 13.74 per meg (units defined in Methods) between 1991 and 2016. ΔAPOOBS is driven by four primary contributors, illustrated in Fig. 2:

$${\Delta {\rm{APO}}}_{{\rm{OBS}}}={\Delta {\rm{APO}}}_{{\rm{FF}}}+{\Delta {\rm{APO}}}_{{\rm{Cant}}}+{\Delta {\rm{APO}}}_{{\rm{AtmD}}}+{\Delta {\rm{APO}}}_{{\rm{Climate}}}$$

where ΔAPOFF is the decrease in APO caused by industrial processes (fossil fuel burning and cement production) which in aggregate consume more than 1.05 moles of O2 for each mole of CO2 released; ΔAPOCant accounts for the oceanic uptake of excess anthropogenic atmospheric CO2; APOAtmD accounts for for air-sea exchanges driven by ocean fertilization from anthropogenic aerosol deposition (increased fertilization leads to increased photosynthesis, with a concomitant release of O2 and uptake of CO2); and ΔAPOClimate accounts for air-sea fluxes of O2, CO2 and N2 driven by ocean processes, including warming-induced changes in solubility, in ocean circulation, and in photosynthesis and respiration (N2 influences O2/N2 ratios). Here, we derive ΔAPOClimate from Eq. (1) and show that it tracks ocean warming.

Figure 2
figure 2

Processes contributing to observed changes in atmospheric potential oxygen (ΔAPOOBS). Industrial processes (fossil fuel burning and cement production, ΔAPOFF) and ocean sink for anthropogenic carbon (ΔAPOCant) remove APO from the atmosphere. The fertilization effect of anthropogenic aerosol deposition which promotes marine photosynthesis (ΔAPOAtmD) and the changes in solubility, biology and ocean circulation due to warming (ΔAPOClimate) release APO into the atmosphere. This study shows that ΔAPOClimate can be used to estimate long term changes in global ocean warming.

We estimate ΔAPOFF using fossil fuel and cement inventories21, finding ΔAPOFF = −142.38 ± 7.65 per meg (Fig. 3). ΔAPOCant is controlled by the increase in atmospheric CO2 and by ocean mixing, which is quantified by transient tracers distributions including chlorofluocarbons (CFCs)22; we find that ΔAPOCant = −147.75 ± 3.69 per meg. ΔAPOCant is relatively precise because it excludes the effect of changing ocean biology and circulation on natural carbon fluxes that are included in ΔAPOClimate. ΔAPOAtmD is derived from ocean model simulations with and without aerosols fertilization (phosphate, iron and nitrogen; Fig. S1)23. ΔAPOAtmD is uncertain, owing in part to uncertainties in iron availability to photosynthetic organisms, but is relatively small compared with the other terms: ΔAPOAtmD = 7.00 ± 3.50 per meg. From Eq. (1), we therefore find ΔAPOClimate = 27.21 ± 16.85 per meg, corresponding to a least-squares linear trend of +1.11 ± 0.68 per meg per year – larger than the trends expected from 26-year natural variations alone in four earth system models (the Community Earth System Model (CESM) and the Geophysical Fluid Dynamics Laboratory (GFDL), Institut Pierre Simon Laplace (IPSL) and University of Victoria (UVic) models). As show in Fig. 3, a clear increase in ΔAPOClimate emerges over the period January 1991 to the end of December 2016.

Figure 3
figure 3

Data-based estimates of global ΔAPOClimate. (a) ΔAPOClimate estimated from observed APO (ΔAPOOBS) from the Scripps Institution of Oceanography network (1991–2016), and corrected from fossil fuel burning, ocean anthropogenic carbon uptake and anthropogenic aerosols deposition (ΔAPOClimate = ΔAPOOBS –ΔAPOFF –ΔAPOCant –ΔAPOAtmD) and their 1σ uncertainty ranges. (b) The increase in global ΔAPOClimate (±1σ interval) exceeds the range of 26-year trends expected from natural variations in four earth system models (CESM, GFDL, IPSL and UVic, shown in grey). Uncertainties and contributions to ΔAPOClimate are given in Tables S3, S4 and S6.

Linking atmospheric potential oxygen to ocean warming

A starting point for understanding ΔAPOClimate is to imagine that O2 and CO2 behave like inert gases, such that the air-sea fluxes are dominated by temperature-driven solubility changes. In this case, APO would increase by around 0.8 per meg per 1022 J of warming, with O2 and CO2 solubility changes accounting for an increase of +1.0 per meg per 1022 J partly offset by the N2 contribution of −0.2 per meg per 1022 J (Methods). Support for the dominance of solubility in ΔAPOClimate can be found in the natural distribution of O2 and carbon in the ocean. Ocean potential oxygen (OPO) is a dissolved tracer that mirrors APOClimate and tracks changes in air-sea O2 and CO2 fluxes17. Observed OPO abundance is strongly tied to ocean potential temperature (Fig. 4): warming induces OPO loss and cooling induces OPO gain. The OPO-to-temperature trend of −4.43 nmol J−1 is within 18% of the trend of −3.65 nmol J−1 expected from solubility alone (OPOsat-to-temperature). Biological effects (related to changes in ocean circulation and photosynthesis/respiration) on CO2 and O2 substantially cancel in OPO (Fig. S2), while thermal impacts reinforce each other, with warming waters releasing both O2 and CO2 to the atmosphere and increasing ΔAPOClimate.

Figure 4
figure 4

Observed link between potential oxygen and ocean heat. OPO concentrations in-situ (OPO, yellow) and at saturation based on O2 and CO2 solubility (OPOsat, grey) as a function of ocean temperature in the GLODAPv2 database49.

Further support for the dominance of solubility on ΔAPOClimate is found on multidecadal timescales in four Earth system models mentioned above, which yield OPO-to-temperature ratios between −4.69 and −4.36 nmol J−1, bracketing the ratio of −4.43 nmol J−1 found in hydrographic observations (Fig. S3). The models also simulate a very close relationship between ΔAPOClimate and the change in global ocean heat content (ΔOHC) that occurs during the simulations (1920–2100), with an atmospheric build-up in APO between 0.82 and 0.98 per meg per 1022 J (Figs. S3 and S4) – close to the ratio expected from temperature-driven solubility changes alone (0.8 per meg per 1022 J). By dividing the simulated APO change into separate biological and thermal components, we show that solubility changes account for more than 80% of ΔAPOClimate, while biologically driven changes account for 5% to 20% (Fig. S4). This partitioning found in response to transient warming is very similar to the partitioning found in hydrographic data (where solubility and biology contribute 82% and 18% respectively, to the OPO-to-temperature ratio, Fig. 4).

Small differences between individual model ΔAPOClimate-to-ΔOHC relationships (0.82 to 0.98 per meg per 1022 J) reflect systematic differences in biological fluxes. Models with stronger biological effects (IPSL and UVic) yield stronger oceanic loss of OPO and stronger release of APO for a given ocean warming (more negative OPO-to-temperature and higher ∆APOClimate-to-∆OHC, Fig. S3b). Using this relationship, we find that a ∆APOClimate-to-∆OHC ratio of 0.86 ± 0.03 per meg per 1022 J is compatible with observed OPO-to-temperature ratio. Combining this constrained ΔAPOClimate-to-ΔOHC ratio (0.86 ± 0.03 per meg per 1022 J) with the observation-based trend in ΔAPOClimate (1.11 ± 0.68 per meg/y), yields a warming trend of 1.29 ± 0.79 × 1022 J yr−1 between 1991 and 2016. As shown in Fig. 1, this APO-based estimate of ocean heat uptake is centered near the high end of recent ocean-based estimates, while agreeing with all estimates to within the uncertainties. In our estimate, the largest single source of uncertainty is the scale error from the span calibration of the O2/N2 analyzer (Table S6), which can be reduced via within-lab and inter-lab comparisons. The uncertainty in the estimated heat uptake since 1991 should therefore be significantly reduced in future updates.


Most recent ocean-based estimates of warming suggest an increase relative to prior estimates1,5,9. The independent APO estimate is fully in line with this upwards revision, although uncertainties are large. Our central value of ΔOHC would call for a steric sea level rise of 1.53 mm y−1 (see Methods), also in agreement with satellite-derived constraints on thermal expansion, corrected for the fresh water contribution (1.50 ± 0.40 mm y−1)24,25. Our result suggests that the ocean contributes 0.80 ± 0.49 W/m2 to the Earth energy imbalance over the 1991–2016 period (Earth surface of 5.1 × 10 14 m2), with implications for equilibrium climate sensitivity and the climate system lagged response to anthropogenic forcing26,27.

We find that the APO-heat coupling (APOClimate) is most robust on decadal and longer time-scales. Strong cancelation of biological O2 and CO2 fluxes is not expected on all temporal scales. On seasonal time-scales, air-sea O2 fluxes driven by marine photosynthesis are ~8 times larger than those of CO2 due to slow equilibration of CO228. More complex coupling is also possible on interannual time-scales29, such as the weaker lagged air-sea CO2 flux compared to O2 during El Nino events30.

Atmospheric O2 and CO2 measurements have previously been applied to estimate global land and ocean CO2 sinks, but relied on ocean heat content estimates and model-based oceanic O2-to-heat ratios to correct for climate-driven O2 outgassing31,32,33. Here we have reversed this logic, using estimates of other quantities to constrain the ocean heating. Our approach exploits the APO-heat relationship, which is stronger than the O2-heat relationship (See Methods for estimate of climate-driven ocean O2 outgassing based on our results). Further work to constrain the separate contributions of O2 and CO2 to APO is needed to refine estimates of ocean and land carbon sinks using atmospheric O2.

The results presented here were previously published in Nature, 563, 105–108 (2018)34. The paper was subsequently retracted due to the underestimation of the uncertainties. Certain systematic errors were treated as random errors, and the uncertainty in the land oxidative ratio on the APO budget was neglected. These issues have been corrected in the current paper. The correction did not substantially change the central estimate of ocean warming but led to a roughly fourfold increase in uncertainties.


Observed changes in atmospheric potential oxygen (ΔAPOOBS)

A change in atmospheric potential oxygen concentration (in per meg) is defined following19:

$$\Delta (\delta {\rm{A}}{\rm{P}}{\rm{O}})[{\rm{p}}{\rm{e}}{\rm{r}}\,{\rm{m}}{\rm{e}}{\rm{g}}]=\Delta ({\delta {\rm{O}}}_{2}/{{\rm{N}}}_{2})+\frac{{\alpha }_{{\rm{B}}}}{{{\rm{X}}}_{{\rm{O}}2}}\cdot ({{\rm{X}}}_{{\rm{C}}{\rm{O}}2}-350)$$


$$({{\rm{\delta }}{\rm{O}}}_{2}/{{\rm{N}}}_{2})[{\rm{per}}\,{\rm{meg}}]=\frac{{{\rm{O}}}_{2}/{{\rm{N}}}_{2}({\rm{sample}})}{{{\rm{O}}}_{2}/{{\rm{N}}}_{2}({\rm{reference}})}\,-\,1$$

where Δ(δO2/N2) is the atmospheric change in δO2/N2 ratios (in per meg), XCO2 is the CO2 concentration in the air parcel (in ppm, i.e. μmol mol−1) and 350 is an arbitrary reference, αB (=1.05) is the approximate O2:CO2 ratio of terrestrial ecosystems20, and XO2 (=0.2094) is the reference value of atmospheric mole fraction of O2 necessary to convert XCO2 from ppm to per meg units. We use an oxidative ratio αB of 1.05 rather than 1.1, following the recommendation of ref. 20. who argue that woody biomass, which has an oxidative ratio near 1.05, likely dominates the long-term carbon sources and sinks on land.

ΔAPOOBS is computed from in situ atmospheric changes in CO2 concentrations and O2/N2 ratios20 measured at stations of the Scripps Institution of Oceanography network (available online at,35. The global average ΔAPOOBS is based on a weighted average of the three stations with the longest record (1991 to 2016), i.e. La Jolla (32.9°N, 117°W), Alert (82.5°N, 62.5°W) and Cape Grim (40.5°S, 144.5°E), with weights of 0.25, 0.25 and 0.5 respectively36. Station’s annual means are based on bi-monthly data fit to a four-harmonic seasonal cycle and a stiff long-term trend18. The uncertainty on ΔAPOOBS was computed by generating 106 time-series with a noise scaled to the random and systematic errors of APO data detailed in Table S3. The uncertainty is taken as the 1-sigma interval (±1 standard deviation) from these 106 realizations (Fig. 3). To assess whether the weighted average of the 3 stations accurately reflects the global trend, we also compared other combinations of stations from 1999 onward when we have reliable data from 9 stations. The differences were consistently smaller than 0.05 per meg yr−1 which is negligible compared to the other combined measurement uncertainties.

Fossil fuel burning and cement production influence on APO (ΔAPOFF)

ΔAPOFF is estimated using annual CO2 emissions from oil, coal, gas, flaring and cement production (ΔCO2(i) in moles)21 weighted by their O2:C combustion ratios Ri18:

$${\Delta {\rm{APO}}}_{{\rm{FF}}}[{\rm{per}}\,{\rm{meg}}]=\sum _{{\rm{i}}}\,\frac{{{\rm{\alpha }}}_{{\rm{B}}}-{{\rm{R}}}_{{\rm{i}}}}{{{\rm{X}}}_{{\rm{O}}2}}\times \frac{{\Delta {\rm{CO}}}_{2({\rm{i}})}}{{{\rm{M}}}_{{\rm{air}}}}$$

where Mair is the number of moles of dry air in the atmosphere (convert moles of CO2 to ppm).

The uncertainty on ΔAPOFF includes uncertainties in CO2 emissions (ΔCO2(i))37 and in combustion ratios (Ri in Tab S3)18. Uncertainties in ΔCO2(i) and Ri are considered not independent in time and were estimated using an autoregressive model with a correlation time-scale of 20 years following the method of ref. 38 (1000 realizations using Monte-Carlo approach). The uncertainty on ΔAPOFF was then estimated by combining the 1000 realizations of ΔCO2(i) and the 1000 realizations of Ri, yielding a set of 106 estimates of ΔAPOFF.

Ocean anthropogenic carbon uptake influence on APO (ΔAPOCant)

We can divide the ocean CO2 uptake (ΔCO2) as the sum of three contributions:

$${\Delta {\rm{C}}{\rm{O}}}_{2}={\Delta {\rm{C}}{\rm{a}}{\rm{n}}{\rm{t}}}_{0}+\Delta {\rm{C}}{\rm{a}}{\rm{n}}{\rm{t}}^{\prime} +{\Delta {\rm{C}}{\rm{O}}}_{2{\rm{C}}{\rm{l}}{\rm{i}}{\rm{m}}{\rm{a}}{\rm{t}}{\rm{e}}}\,({\rm{S}}1)$$

where ΔCant0 is the flux driven by the rise in CO2 assuming steady ocean circulation (ΔCant0 is negative, corresponding to uptake by the ocean), ΔCO2Climate is the flux driven by the action of climate on natural carbon in the ocean (ΔCO2Climate is positive, i.e. warming reduces the uptake of natural carbon), and ΔCant’ is the remainder, which accounts for impact of circulation changes on the uptake of carbon driven by rising CO2 (ΔCant’ is positive, i.e. warming reduces the uptake of Cant). ΔAPOCant can be expressed as the weighted sum of the two terms ΔCant0 and ΔCant’:

$${\Delta {\rm{APO}}}_{{\rm{Cant}}}[{\rm{per}}\,{\rm{meg}}]=\frac{{{\rm{\alpha }}}_{{\rm{B}}}}{{{\rm{X}}}_{{\rm{O}}2}\times {{\rm{M}}}_{{\rm{air}}}}\times {(\Delta {\rm{Cant}}}_{0}+\Delta {\rm{Can}}{\rm{t}}^{\prime} )$$

where ΔCant0 and ΔCant’ are in moles. Note that ΔCO2Climate is accounted for in ΔAPOClimate.

ΔCant0 is taken from the recent ocean inversion scheme with assimilation of observed potential temperature, salinity, radiocarbon, and CFC-11 of ref. 22 updated to 2016. ΔCant’ cannot be derived from observations and was estimated to be 0.05 PgC y−1, equivalent to a trend of +0.11 per meg−1, using model simulations (see details in section Model-based ΔCant’ below).

The uncertainty on ΔAPOCant is related to uncertainties in ΔCant0 and ΔCant’. We allow for uncertainty in ΔCant0 following ref. 22 using the 10 sensitivity experiments (on ocean vertical and isopycnal diffusivities, data constraint, gas exchange coefficient etc.) available for the ocean inversion and an estimate of the interannual variability in the ocean sink of a 0.2 PgC y−1. We also allow an additional 1% uncertainty (<0.03 PgC/y) in ΔCant0 due to imperfectly known atmospheric CO2 history39, taking account of sensitivity to start date (1765 vs 1791), to degree of temporal smoothing, and to using different versions of the record since 1958 (Mauna Loa record versus average of Mauna Loa and South Pole records). This estimate used a variant of the box-diffusion model40, and CO2 data from ref. 41 and the Scripps CO2 program ( Uncertainties on ΔCant’ are assumed to be 100% of the model-based estimate of ΔCant’ (see details in section Model-based ΔCant’ below).

Ocean fertilization and atmospheric deposition of anthropogenic aerosol (ΔAPOAtmD)

Deposition of anthropogenic aerosol from fossil fuel, biomass burning etc. fertilizes the ocean with nutrients, increases surface photosynthesis and sub-surface respiration42,43,44. The effect of aerosol fertilization is partly counterbalanced by biological processes such as a decline in nitrogen fixation, which would be immediate, and an increase in denitrification in the water column, which would be on time-scales of several 100 years45. Fixed anthropogenic nitrogen also fertilizes the land biosphere and coastal oceans by river runoffs, but in these cases, efficient denitrification returns fixed nitrogen to the atmosphere and has little impact on the APO budget on the decadal timescales considered here. The impact of anthropogenic aerosol on O2, CO2 and APO air-sea fluxes is evaluated with the IPSL ocean model NEMO-PISCES v246 using the difference between simulations with aerosols and a simulation in which the aerosol deposition is fixed to a constant pre-industrial value (equivalent to year 1850, Fig. S1)23. We use four simulations with varying aerosols: one includes the combined effect of nitrogen (N), iron (Fe) and phosphorus (P) aerosol deposition, whereas the 3 others only include their individual contribution (N-only, Fe-only or P-only, Fig. S1 and Table S5). Uncertainties at 1-sigma level on ΔAPOAtmD are assumed to be ±50%. See Table S4.

Combined, N, Fe and P deposition accounts for an O2 outgassing of 19.0 Tmol y−1 for the 1980/2007 period (16 Tmol y−1 for entire 1960/2007 simulation period) and an oceanic CO2 uptake of 8.3 Tmoly−1 for the 1980–2007 period (6.8 Tmo ly−1 for entire 1960–2007 simulation period, Fig. S1 and Table S5). The overall impact is ΔAPOAtmD = + 0.28 per meg y−1 over 27 years of simulation (1980 to 2007), which we extrapolate to our 1991 to 2016 period. Increased O2 outgassing accounts for an increase in APO of +0.51 per meg y−1, and CO2 uptake accounts for a change in APO of −0.23 per meg y−1 (APOAtmD(O2) and APOAtmD(CO2) in Table S3).

The overall effect of N, Fe and P is smaller than the sum of the individual effects (Fig. S1), because of the interplay between the aerosol deposition pattern and nutrient co-limitations in the ocean. Phytoplankton growth in the ocean depends on the availability of the most limiting nutrient. While more available N will promote photosynthesis in regions where N is limiting (for example the tropical Atlantic Ocean), the effect is negligible in regions where Fe, P or any other nutrient are limiting (for example the Southern Ocean) (see Fig. 2 in ref. 23).

To our knowledge this is the first estimate of the impact of anthropogenic aerosol deposition on both O2 and CO2 air-sea fluxes at the global scale. Note however that ref. 18 used anthropogenic aerosol N inventories and scaling arguments to estimate an ocean O2 loss due to anthropogenic N-deposition only of ~10 ± 10 Tmol y−1, slightly weaker than our model estimate of 15.5 Tmol y−1.

ΔAPOClimate trends and uncertainty analysis

We compute the APO response to climate change (ΔAPOClimate) via

$${\Delta {\rm{APO}}}_{{\rm{Climate}}}={\Delta {\rm{APO}}}_{{\rm{OBS}}}-{\Delta {\rm{APO}}}_{{\rm{FF}}}-{\Delta {\rm{APO}}}_{{\rm{Cant}}}-{\Delta {\rm{APO}}}_{{\rm{AtmD}}}$$

We combine the estimates of ΔAPOFF, ΔAPOCant and ΔAPOAtmD plus estimates of the contribution from variations in the oxidative ratio to obtain 106 time-series of ΔAPOFF + ΔAPOCant + ΔAPOAtmD and obtain 106 time-series of ΔAPOClimate using the 106 time-series of ΔAPOOBS. We computed the ΔAPOClimate trend and its uncertainty based on the distribution of the unweighted least square fits to each of the 106 ensemble realizations of ΔAPOClimate generated by combining all sources of uncertainty. We find a ΔAPOClimate trend of 1.11 ± 0.68 per meg yr−1 for 1991–2016. The individual contributions to the uncertainty are shown in Table S6.

Hydrography-based estimates of ocean heat uptake (ΔOHC)

We used four global-ocean estimates of ΔOHC based on hydrographic measurements in Fig. 1. Surface to 2000 m warming rates are from ref. 6 (PMEL), ref. 5 (MRI,, an updated version of ref. 47 (NCEI,, and ref. 8 (CHEN,, with the revised deep ocean (depth >2000 m) constant linear warming rate of 0.10 ± 0.03 × 1022 J yr−1 of ref. 7 based on the global ship-based sections program (GO-SHIP;

Ocean observations of ocean potential oxygen (OPO)

We used in-situ ocean observations from GLODAPv249 combined with an anthropogenic carbon estimate22 interpolated at the location of each sample to compute 78,456 values (Glodapv2 quality control = 0, marginal seas and coastal waters were removed) of Oceanic Potential Oxygen (OPO)17 via

$${\rm{O}}{\rm{P}}{\rm{O}}={{\rm{O}}}_{2}+{\alpha }_{{\rm{B}}}\times {{\rm{C}}}_{{\rm{p}}{\rm{i}}}^{\ast }$$

where O2* and Cpi* are the ocean conservative tracers related to air-sea fluxes of O2 and pre-industrial carbon50, and αB is the terrestrial oxidative ratio (αB = 1.05). The thermal component (solubility-driven) of OPO (OPOsat) is computed as:

$${{\rm{OPO}}}_{{\rm{sat}}}={{\rm{O}}}_{2{\rm{sat}}}+{{\rm{\alpha }}}_{{\rm{B}}}\times {{\rm{C}}}_{{\rm{pisat}}}$$

where O2sat is the dissolved O2 concentration at saturation with the observed temperature and salinity51 and Cpisat is the dissolved inorganic carbon concentration expected at the observed temperature and salinity, and assuming equilibrium with a pre-industrial partial pressure of CO2 of 280 ppm and using pre-formed alkalinity52.

Solubility-driven changes in oceanic and atmospheric potential oxygen

Figure S2 shows a tight and quasi-linear link between observed OPO and potential temperature (−4.43 nmol/J, r2 = 0.95), similar to the link found between OPOsat and potential temperature (−3.65 nmol/J, r2 = 0.93). This suggests that changes in OPO and hence ΔAPOClimate are driven primarily by changes in thermal air-sea fluxes. In these observations, departures of dissolved oxygen and carbon concentrations (O2* and Cpi*) from their respective saturation curves (O2sat and Cpisat) due to biological activity tend to balance (Fig. S2). By contrast, thermal effects reinforce each other (O2sat and Cpisat both decrease with increasing temperature) and biological effects compensate each other (O2* > O2sat and Cpi* < Cpisat).

Changes in APO expected from changes in gas solubility in the ocean is an increase of 2.95 nmol per J of warming, which includes the outgassing of O2 and CO2 following OPOsat (3.65 nmol/J) and the release of of N2 (0.6 nmol/J) (Fig. S2b). A change of 2.95 nmol per J of warming is equivalent to an increase of 0.8 per meg/1022 J of APO in the atmosphere (=(2.95 × 10−9)/(3.7 × 1019) × 1022 = 1.0 × 10−6 = 0.8 per meg per 1022 J, with 3.7 × 1019 the number of moles of O2 in the atmosphere). O2 and CO2 solubility alone yield an increase in APO of 1.2 per meg/1022 J, which is partly counterbalanced by the outgassing of N2 that decreases APO by 0.2 per meg/1022 J (via the increase of the O2/N2 ratio).

Earth system model experiments

We used 4 Earth-system models (ESMs): the Geophysical Fluid Dynamics Laboratory Earth System Models with a nominally level vertical coordinate version GFDL-ESM2M (called GFDL here)53,54, the Institut Pierre-Simon Laplace Coupled Model 5 version IPSL-CM5A-LR (IPSL here)55, the Community Earth System Model large ensemble CESM-LE (CESM here)56 and the UVic model version 2.9 (UVic here)57. Evaluation of these models and their biogeochemical components can be found in previous studies54,56,58,59,60. GFDL, IPSL and UVic participated in the Coupled Model Intercomparison Project Phase 5 (CMIP5)61.

For GFDL, IPSL and UVic, we used the CMIP5 business as usual “historical-RCP8.5” scenario, the feedback experiment “esmFdbk3” that only includes warming-driven changes associated with anthropogenic emissions (e.g. radiation effects) and the fixed-climate experiment “esmFixClim3” that only includes the direct biogeochemical effects of increasing atmospheric CO2 (e.g. uptake of anthropogenic carbon, acidification etc.). For CESM, we also used the historical and RCP8.5 experiments and the separation between anthropogenic carbon from the natural carbon available in this model (carbon tracer separation approach). The feedback approach used for GFDL, IPSL and UVic removes all direct biogeochemical effects of rising atmospheric CO2 on the air-sea O2 and CO2 exchanges, whereas the natural carbon tracer separation approach used for CESM still includes the biogeochemical impacts of increasing atmospheric CO2 on the carbon cycle (e.g. acidification) even while it excludes the anthropogenic carbon itself. However, we expect this effect to be small and negligible on our results.

We also used the multicentury preindustrial control simulation “piControl” with no increase in atmospheric CO2 to correct for model drift and to estimate the natural internal variability of ΔAPOClimate (Fig. 2). We used model results over the 1920–2100 period, which were available for the four models.

Model OPO was computed as in the observations. Note that for CESM we removed subsurface regions of high denitrification in the eastern equatorial Pacific and Bay of Bengal where oxygen and O2* in this model have unrealistic values62.

Model anthropogenic ΔCant’

The component ΔCant’ was derived from Eq. (S1) (ΔCant’ = ΔCO2 − ΔCant0 − ΔCO2Climate) using CMIP5 model simulations. ΔCO2 was taken from experiment RCP8.5, ΔCant0 from experiment esmFixClim3, and ΔCO2Climate from experiment esmFdbk3. Note that the control simulation was also used to correct model drift. We estimated ΔCant’ = 0.05 ± 0.05 PgC yr−1 for the 1991 to 2016, based on the results of the three models, which individually yields ΔCant’ of 0.0 PgC yr−1 (IPSL), 0.11 PgC yr−1 (GFDL) and 0.11 PgC yr−1 (UVic), and assuming an uncertainty of ±100%. This corresponds to a trend of 0.11 ± 0.11 per meg yr−1.

Model ΔAPOClimate to ΔOHC ratios and uncertainty

Model ΔAPOClimate is computed using individual contributions from O2, CO2 and N2 via:

$$\begin{array}{c}{\Delta {\rm{APO}}}_{{\rm{Climate}}}[{\rm{per}}\,{\rm{meg}}]={\Delta {\rm{APO}}}_{({\rm{O}}2)}+{\Delta {\rm{APO}}}_{({\rm{CO}}2)}+{\Delta {\rm{APO}}}_{({\rm{N}}2)}\\ {\Delta {\rm{APO}}}_{{\rm{Climate}}}[{\rm{per}}\,{\rm{meg}}]=\frac{1}{{{\rm{X}}}_{{\rm{O}}2}\,{{\rm{M}}}_{{\rm{air}}}}\times ({\Delta {\rm{F}}}_{{\rm{O}}2}+{{\rm{\alpha }}}_{{\rm{B}}}\times {\Delta {\rm{F}}}_{{\rm{CO}}2}-\frac{{{\rm{X}}}_{{\rm{O}}2}}{{{\rm{X}}}_{{\rm{N}}2}}\times {\Delta {\rm{F}}}_{{\rm{N}}2})\end{array}$$

where ΔFO2, ΔFCO2 and ΔFN2 are the changes in air-sea fluxes of O2, CO2 and N2 respectively (in moles), Mair is the number of moles of dry air in the atmosphere and XN2 and XO2 are the reference atmospheric mixing ratio of N2 and O2 respectively63. O2 and CO2 fluxes are simulated in the models. N2 air-sea fluxes, which impact the O2 atmospheric mixing ratio (because O2 is ~20% of the atmospheric composition), are quantified from the global ocean temporal changes in N2 solubility computed from model changes in temperature and salinity64.

The link between long-term changes in APOClimate and ocean heat content, i.e. ΔAPOClimate-to-ΔOHC ratios, were computed for each model using the 180 years of simulations (1920 to 2100). Resulting ΔAPOClimate-to-ΔOHC ratios vary between 0.82 and 0.98 per meg per 1022 J of warming (Fig. S3). These ratios include uncertainty in the natural climate variations at interannual and decadal time-scales and uncertainty in the O2:C oxidative ratio associated with global gains and losses of O2 and CO2 by terrestrial ecosystems (αB = 1.05 ± 0.05). The uncertainty due to interannual variations was evaluated by computing ΔAPOClimate-to-ΔOHC ratios using multiple 26 year-long segments from the 180 year simulations. We obtained 616 ΔAPOClimate-to-ΔOHC ratios (154 time-series of 26 years per model) and used the standard deviation between these ratios as a measure of the uncertainty.

We combine results from all models along with modeled and observed hydrographic relationships between OPO and potential temperature to establish an observationally-constrained estimate of the ΔAPOClimate-to-ΔOHC ratio, as shown in Fig. S3b. We first carry out this procedure with a fixed oxidative ratio αB of 1.05, which yields ΔAPOClimate-to-ΔOHC = 0.86 ± 0.03 per mer per 1022 J. Repeating the same procedure using oxidative ratios αB of 1.0 and 1.1 applied to both APO and OPO yield differences in ΔAPOClimate-to-ΔOHC smaller than 0.01 per meg per 1022J, showing that sensitivity to the oxidative ratio is very small. Note that on Fig. S3b, the model error bars include two contributions to the uncertainties on the simulated ΔAPOClimate-to-ΔOHC ratios (interannual variations and O2:C ratio) that combine to yield ±0.01 per meg per 1022 J for the CESM and GFDL models, ±0.02 per meg per 1022 J for the UVic model and ±0.05 per meg per 1022 J for the IPSL model (1-sigma).

Steric component of sea level rise

We evaluated the steric component of sea level rise associated with a OHC of 1.3 × 1022 J yr−1 to be 1.53 mm y−1. Following ref. 65, this calculation assumes that 45% of the warming occurs below 700 m and that the steric rise is 1 mm per 0.60 × 1022 J above 700 m and 1 mm per 1.15 × 1022 J below 700 m (i.e. global steric rise of 1 mm per 0.84 × 1022 J). Assuming that 48% of the warming occurs below 700 m (ref. 6) would yield a global steric rise of 1 mm per 0.86 × 1022 J and change our estimate by less than 3%. Our estimate is also consistent with the recent hydrography-based estimate of the WCRP Global Sea Level Budget Group66.

Link to global ocean de-oxygenation

Our application of O2 atmospheric measurements to constrain long-term ocean warming can be compared with earlier work considering warming-driven oceanic O2 outgassing. Multiplying our warming rate of 1.29 ± 0.79 × 1022 J yr−1 by the O2-to-heat ratios simulated by the four ESMs (−3.70 ± 0.80 nmol O2 J−1), yields an ocean loss of 48 ± 30 Tmol O2 y−1. Adding a loss of ~19 Tmol O2 y−1 due to anthropogenic aerosols (Table S5), yields a global ocean outgassing of 67 ± 35 Tmol O2 y−1, in the range of previous estimates based on atmospheric data67 (~40 Tmol O2 y−1), ocean data above 1000 m68,69 (55–65 Tmol O2 y−1) and global ocean data70 (96 ± 42 Tmol O2 y−1). This calculation implies that ocean CO2 uptake is reduced by warming at a ratio of ~0.70 nmol of CO2 per Joule (difference between O2-to-heat ratio of 3.70 nmol J−1 and OPO-to-heat ratio of 4.43 nmol J−1).